So I found a maybe useless multiclass build that seems fairly niche since its so obvious and MAD that people might pass over it automatically: a Bladesinger/Hexblade.
This does require you to have good Charisma and good Intelligence, but it works surprisingly well thanks to one little thing: RAW, the Bladesinger's Extra Attack states that you can cast 'one of your cantrips' in place of an attack, not just a wizard cantrip. Ergo, if you take Hexblade with Eldritch Blast and Agonizing Blast, you functionally have three attacks by combined level 8, with two of those having an 120ft range, not to mention the various buffs warlocks can add to EB with Invocations. This also mitigates one of the weaknesses of Hexblade: the fact that its always better to just Eldritch Blast rather than get up close and attack. Paired with Haste, Bladesong and Hexblade's Curse, a 20th level half-elf Bladesinger/Hexblade built using Standard Array (15 Int, 14 Cha, 13 Dex, 12 Con, 10 Wis, 8 Str) and using a longsword can deal a grand total of: 3d10+2d8+40 damage (an average of 68 damage) after expending one Bladesong, a 3rd level slot and Hexblade's Curse. They also do this whilst having a +5 to AC and the possibility of a multitude of other buff spells.
An issue with this is that Eldritch Blast is a ranged attack and is therefore at disadvantage if there is an enemy within 5 feet of you. You can avoid this by taking crossbow expert but with a build this made you might want to get every ASI you can
An alternative would be to have a weapon attack with a range of more than 5 feet, there are a few ways of doing this but it does limit the options for the build. However if if an enemy moves within 5ft you are bake to being at disadvantage on eldritch blast unless you move away (either risking an op attack or burning a misty step spell slot).
Bladesinger ends if you attack with a weapon using 2 hands, so if you use a weapon with reach you are limited to whip, or a lance if you ar emounted (if Tasha's customization is allowed you could get this by being a wood elf or something and swapping out the weapon proficiency).
I cannot see a reason why hand crossbow doesn't work, though you will probably want to discuss with the DM about this and the feats you might need. You need a free hand to load and while you can cast eldritch blast without components or a focus there may be issues if you want ot cast spells that do have an M component. (You might be able to work through this with object interactions taking warcaster will eliminate the issue.
You could be of a race that is long limbed such as bugbear so can attack with a non-reach weapon from 10 ft away.
So I found a maybe useless multiclass build that seems fairly niche since its so obvious and MAD that people might pass over it automatically: a Bladesinger/Hexblade.
This does require you to have good Charisma and good Intelligence, but it works surprisingly well thanks to one little thing: RAW, the Bladesinger's Extra Attack states that you can cast 'one of your cantrips' in place of an attack, not just a wizard cantrip. Ergo, if you take Hexblade with Eldritch Blast and Agonizing Blast, you functionally have three attacks by combined level 8, with two of those having an 120ft range, not to mention the various buffs warlocks can add to EB with Invocations. This also mitigates one of the weaknesses of Hexblade: the fact that its always better to just Eldritch Blast rather than get up close and attack. Paired with Haste, Bladesong and Hexblade's Curse, a 20th level half-elf Bladesinger/Hexblade built using Standard Array (15 Int, 14 Cha, 13 Dex, 12 Con, 10 Wis, 8 Str) and using a longsword can deal a grand total of: 3d10+2d8+40 damage (an average of 68 damage) after expending one Bladesong, a 3rd level slot and Hexblade's Curse. They also do this whilst having a +5 to AC and the possibility of a multitude of other buff spells.
20th level- EB is 4d10 with 4 beams. Cantrips scale off of character level, not class level. Agonizing blast would be per beam and you should have enough ASIs to bump your charisma to 20 since it will be your attack stat 4×5=20 from AB. 4d10+20 so far.
Longsword can be held with two hands and hex warrior only excludes the two-handed property, not the versatile property. Hex warrior can be used with a longsword two handed meaning that you could get 2 attacks at 1d10+5 if you are hated. 2d10+10.
Hexblade's Curse gives a damage bonus equal to your proficiency bonus. You've got 6 attacks and a proficiency bonus of 6, 36 points. This means that max damage would be 6d10+66 (99 average) as long as you didn't want to keep Bladesong up. It drops to 4d10+2d8+66 with Bladesong active dropping to 97 average damage. If you go at least 14 levels into wizard, you can add your intelligence modifier to your melee attacks while Bladesong is active, giving you 10 more damage and 107 average damage potential.
Assuming a base Studded Leather Armor and a starting dex of 14 (13 +1 from half elf) you'd have a 14 AC. If you leveled wizard 8/warlock 12, wizard 12/warlock 8, wizard 16/Warlock 4, or Wizard 19/Warlock 1 (which would lose AB and thus lower the damage from charisma) and used two ASIs on charisma, you could max dexterity with the other 3 to have a base 17 AC which is the same as Half-Plate with a 14 dex but works with Bladesong. However, you'd probably want to max intelligence, leaving 1 ASI at most for dex leaving your base AC at either 14 or 15. With Bladesong, you gain a bonus to AC equal to your intelligence modifier, giving you a +3 to +5. Let's assume that you leveled at least 4 in Warlock and did so in a way to get all 5 possible ASIs from this build, bumping charisma and intelligence to max with 4 of them and dex to 16 with the other. 15 base +5 bonus AC from Bladesong +2 from haste is a respectable 22 AC, with Shield in your pocket to bump it further to 27. It might be better to stay ranged when not in bladesong and bump constitution instead of dexterity.
Concentration saves are remarkably solid with Eldritch Mind and Bladesong. A solid alternate would be to get Resilient (Con) instead of +2 Constitution, but that would also drop your HP total by 20.
An issue with this is that Eldritch Blast is a ranged attack and is therefore at disadvantage if there is an enemy within 5 feet of you. You can avoid this by taking crossbow expert but with a build this made you might want to get every ASI you can
An alternative would be to have a weapon attack with a range of more than 5 feet, there are a few ways of doing this but it does limit the options for the build. However if if an enemy moves within 5ft you are bake to being at disadvantage on eldritch blast unless you move away (either risking an op attack or burning a misty step spell slot).
Bladesinger ends if you attack with a weapon using 2 hands, so if you use a weapon with reach you are limited to whip, or a lance if you ar emounted (if Tasha's customization is allowed you could get this by being a wood elf or something and swapping out the weapon proficiency).
I cannot see a reason why hand crossbow doesn't work, though you will probably want to discuss with the DM about this and the feats you might need. You need a free hand to load and while you can cast eldritch blast without components or a focus there may be issues if you want ot cast spells that do have an M component. (You might be able to work through this with object interactions taking warcaster will eliminate the issue.
You could be of a race that is long limbed such as bugbear so can attack with a non-reach weapon from 10 ft away.
If you go hand crossbow, the 14th level bladesinger feature is off the table. You could swing to a wizard 6/Warlock 14 build to pick up Lifedrinker to make up for it. You would then have 6 invocations to play with and three spoken for with Agonizing Blast, Eldritch Mind, and Lifedrinker. Lance of Lethargy, Grasp of Hadar, and Repelling Blast would all give your EB some control and help keep your hand crossbow within range of a target while limiting your total exposure. I'd only go with a max of two with Lance of Lethargy likely being the odd one out, but there are enough other invocations that could give a good bonus to give consideration to other aspects. Picking up improved pact weapon and swapping it out once you got a magic weapon would also give a damage bonus or when you pick up lifedrinker.
The other option would be to forgo Eldritch Mind for Devil's Sight, make use of Darkness (either by forgoing Haste or perhaps from a friendly Shadow Monk or Shadow Sorcerer looking for an edge or from any eligible ally looking to control the battlefield) to give your AC a "boost" and limit the risk of OAs. This would likely necessitate the pick up of War Caster for the fifth ASI instead of dexterity or constitution, though Resilient (con) could still be good. War Caster would actually be a good pick up for this build in general.
Longsword can be held with two hands and hex warrior only excludes the two-handed property, not the versatile property. Hex warrior can be used with a longsword two handed meaning that you could get 2 attacks at 1d10+5 if you are hated. 2d10+10.
While that is true bladesong ends " if you use two hands to make an attack with a weapon." so if you want to use a longsword while bladesinging you would be limited to 1d8+5.
If you are using a longsword you also really need crossbow expert or nearly all the agonising blasts will be at disadvantage. An ironic thing with this build is crossbow expert is less important if you use a hand crossbow. As the OP said it is really MAD so I wouldn't really consider it unless I rolled for stats and rolled really well.
I mentioned that it wouldn't work two hands with Bladesong in the next paragraph after the one you quoted. It's only a 2 pt average difference so not a big deal regardless and the bladesinger 14 ability more than makes up for it.
I think I'd rather go Mobile Feat than crossbow expert if I was going to sink a feat into trying to get my EBs to not be cast at disadvantage. Otherwise, I'd just use strategy to limit my exposure to either OAs or stay at range until I was comfortable moving in.
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So I found a maybe useless multiclass build that seems fairly niche since its so obvious and MAD that people might pass over it automatically: a Bladesinger/Hexblade.
This does require you to have good Charisma and good Intelligence, but it works surprisingly well thanks to one little thing: RAW, the Bladesinger's Extra Attack states that you can cast 'one of your cantrips' in place of an attack, not just a wizard cantrip. Ergo, if you take Hexblade with Eldritch Blast and Agonizing Blast, you functionally have three attacks by combined level 8, with two of those having an 120ft range, not to mention the various buffs warlocks can add to EB with Invocations. This also mitigates one of the weaknesses of Hexblade: the fact that its always better to just Eldritch Blast rather than get up close and attack. Paired with Haste, Bladesong and Hexblade's Curse, a 20th level half-elf Bladesinger/Hexblade built using Standard Array (15 Int, 14 Cha, 13 Dex, 12 Con, 10 Wis, 8 Str) and using a longsword can deal a grand total of: 3d10+2d8+40 damage (an average of 68 damage) after expending one Bladesong, a 3rd level slot and Hexblade's Curse. They also do this whilst having a +5 to AC and the possibility of a multitude of other buff spells.
An issue with this is that Eldritch Blast is a ranged attack and is therefore at disadvantage if there is an enemy within 5 feet of you. You can avoid this by taking crossbow expert but with a build this made you might want to get every ASI you can
An alternative would be to have a weapon attack with a range of more than 5 feet, there are a few ways of doing this but it does limit the options for the build. However if if an enemy moves within 5ft you are bake to being at disadvantage on eldritch blast unless you move away (either risking an op attack or burning a misty step spell slot).
20th level- EB is 4d10 with 4 beams. Cantrips scale off of character level, not class level. Agonizing blast would be per beam and you should have enough ASIs to bump your charisma to 20 since it will be your attack stat 4×5=20 from AB. 4d10+20 so far.
Longsword can be held with two hands and hex warrior only excludes the two-handed property, not the versatile property. Hex warrior can be used with a longsword two handed meaning that you could get 2 attacks at 1d10+5 if you are hated. 2d10+10.
Hexblade's Curse gives a damage bonus equal to your proficiency bonus. You've got 6 attacks and a proficiency bonus of 6, 36 points. This means that max damage would be 6d10+66 (99 average) as long as you didn't want to keep Bladesong up. It drops to 4d10+2d8+66 with Bladesong active dropping to 97 average damage. If you go at least 14 levels into wizard, you can add your intelligence modifier to your melee attacks while Bladesong is active, giving you 10 more damage and 107 average damage potential.
Assuming a base Studded Leather Armor and a starting dex of 14 (13 +1 from half elf) you'd have a 14 AC. If you leveled wizard 8/warlock 12, wizard 12/warlock 8, wizard 16/Warlock 4, or Wizard 19/Warlock 1 (which would lose AB and thus lower the damage from charisma) and used two ASIs on charisma, you could max dexterity with the other 3 to have a base 17 AC which is the same as Half-Plate with a 14 dex but works with Bladesong. However, you'd probably want to max intelligence, leaving 1 ASI at most for dex leaving your base AC at either 14 or 15. With Bladesong, you gain a bonus to AC equal to your intelligence modifier, giving you a +3 to +5. Let's assume that you leveled at least 4 in Warlock and did so in a way to get all 5 possible ASIs from this build, bumping charisma and intelligence to max with 4 of them and dex to 16 with the other. 15 base +5 bonus AC from Bladesong +2 from haste is a respectable 22 AC, with Shield in your pocket to bump it further to 27. It might be better to stay ranged when not in bladesong and bump constitution instead of dexterity.
Concentration saves are remarkably solid with Eldritch Mind and Bladesong. A solid alternate would be to get Resilient (Con) instead of +2 Constitution, but that would also drop your HP total by 20.
Consider https://ddb.ac/characters/65340403/chd2A6
If you go hand crossbow, the 14th level bladesinger feature is off the table. You could swing to a wizard 6/Warlock 14 build to pick up Lifedrinker to make up for it. You would then have 6 invocations to play with and three spoken for with Agonizing Blast, Eldritch Mind, and Lifedrinker. Lance of Lethargy, Grasp of Hadar, and Repelling Blast would all give your EB some control and help keep your hand crossbow within range of a target while limiting your total exposure. I'd only go with a max of two with Lance of Lethargy likely being the odd one out, but there are enough other invocations that could give a good bonus to give consideration to other aspects. Picking up improved pact weapon and swapping it out once you got a magic weapon would also give a damage bonus or when you pick up lifedrinker.
The other option would be to forgo Eldritch Mind for Devil's Sight, make use of Darkness (either by forgoing Haste or perhaps from a friendly Shadow Monk or Shadow Sorcerer looking for an edge or from any eligible ally looking to control the battlefield) to give your AC a "boost" and limit the risk of OAs. This would likely necessitate the pick up of War Caster for the fifth ASI instead of dexterity or constitution, though Resilient (con) could still be good. War Caster would actually be a good pick up for this build in general.
While that is true bladesong ends " if you use two hands to make an attack with a weapon." so if you want to use a longsword while bladesinging you would be limited to 1d8+5.
If you are using a longsword you also really need crossbow expert or nearly all the agonising blasts will be at disadvantage. An ironic thing with this build is crossbow expert is less important if you use a hand crossbow. As the OP said it is really MAD so I wouldn't really consider it unless I rolled for stats and rolled really well.
I mentioned that it wouldn't work two hands with Bladesong in the next paragraph after the one you quoted. It's only a 2 pt average difference so not a big deal regardless and the bladesinger 14 ability more than makes up for it.
I think I'd rather go Mobile Feat than crossbow expert if I was going to sink a feat into trying to get my EBs to not be cast at disadvantage. Otherwise, I'd just use strategy to limit my exposure to either OAs or stay at range until I was comfortable moving in.