I haven't done a Warlock subclass before, and was just wanting to know others' opinion on who this subclass is going. I understand that there are places that need to be reworded for the sake of grammar, but I'm primarily looking for how thematic is the expanded spell list, is the class itself balanced and what else needs to be reword, added or removed.

The Ancient Archemage

You have made a pact with a mysterious but powerful caster of ancient times. Your pact with patron grants you a fraction of their knowledge and power; while bolstering your ability to cast spells. They may have given you this pact so that you may gather magical artifacts, learn new secrets of the arcane, or as a pawn in their schemes. Patrons of such power could include any variety of Liches, Ancient Wizard’s, or even the infamous the momentary god, Karsus.

• Expanded Spell List
  • Spell level – 1st – Magic Missile, Identify
  • Spell level – 2nd – Levitate, Nystul’s Magic Aura
  • Spell level – 3rd – Fireball, Lightning Bolt
  • Spell level – 4th – Private Sanctum, Secret Chest
  • Spell level – 5th – Cone of Cold, Teleportation circle
• Level 1
  • Arcane Bolster - At 1st level, you gain 1 additional spell slot, and you gain proficiency in the Arcana skill. Your proficiency bonus is doubled for any ability check you make with the Arcana skill.
  • Tome of Arcane Knowledge - At 1st level, your patron gives you a grimoire named the Tome of Arcane Knowledge. You learn two 1st level spells from the wizard spell list and for you, they count as warlock spells. These spells you can each cast without expending a spell slot once per long rest. If at 3rd level, if you choose the pact of the tome, this grimoire can be treated as the Book of Shadow.
• Level 6
  • Selected studies - Starting at 6th level, you have come to the point where you have found a specialty of magic. You learn two wizard spells of 3rd level or lower, and they are warlock spells for you. These spells you can each cast without expending a spell slot once per long rest.
• Level 10
  • Unrestrained Learning - Beginning at 10th level, you have become more ambitious in learning new spells. You learn two 5th level spells from the wizard spell list. Additionally, you can pick one 1st level wizard spell that can be cast at will. Each wizard spell gained from the feature count as warlock spells to you.
• Level 14
  • Patron’s Secret - Starting at 14th level, you peer into the knowledge of spells in which your patron has learned over the years. You can cast a spell of your choice from the warlock or wizard spell list. The spell must be of level for which you can cast, or at the level of mystic arcanum known. If the spell isn’t a warlock spell, its counts as a warlock spell when you cast it. You can’t use this feature again until you finish a long rest.

I like the idea, I think there's a lot of potential, but there's two spots I worry about.

First is that this subclass feels like there's one right pact boon option and doesn't give room for others. Even Hexblade, a subclass built for pact of the blade still works well with pact of the tome, eldritch blasts and whatnot.

I also worry that in effect, you're handing out two more spell slots, of the highest level they can cast, per day. It feels far too powerful. 

Quote from Astromancer >>

I like the idea, I think there's a lot of potential, but there's two spots I worry about.

First is that this subclass feels like there's one right pact boon option and doesn't give room for others. Even Hexblade, a subclass built for pact of the blade still works well with pact of the tome, eldritch blasts and whatnot.

I also worry that in effect, you're handing out two more spell slots, of the highest level they can cast, per day. It feels far too powerful. 

Yeah Pact Chain and Tome are the definite go to for this subclass, but I also felt shoehorned into pact of the blade for hexblade and even as state there are builds that don't need it. If they chose to go pact of the blade they could make that their arcane focus with the "improved pact weapon" eldritch invocation. So i'll likely change the tome idea into just a shared knowledge from the patron giving the additional spells.

Now I do agree with you that the multiple spells that are once per day free needs to be cut back, and just limited to the 1 additional spell slot and the capstone spell. It just playing my current hex blade, I've always felt like I've always needed just 1 additional spell slot even... even though my hexblade is a frontline tank so having spell slots it isn't much of an issue when I can fix most of my problems with an axe to the face. When I start to think about making a caster warlock, just having that one additional slot I feel would make a big difference.

This sounds cool, but adding an entire spell slot and a a bunch of free spells seems really strong. That being said I think this could be tweaked into something really interesting.

Some thoughts to perhaps make it more balanced.

1st level: Additional wizard cantrip, proficiency in Arcana, a first level wizard spell you can cast once per long rest. 

6th level: An ability dealing force damage, maybe like the War Magic's Power Surge or the Arcana Domain's Spell Breaker.

10th level: That 1st level spell can be cast a number of times equal to your charisma modifier.

14th level: An additional spell slot and a high level spell you can cast once per long rest.

Okay I'll post an update to the sublcass later today with some of the suggestions. Any opinion on the choice spells for the extended spell list?

Okay lets try this again, what do you think of this iteration. The bonus damage idea made me think of doing another pool of dice (unsure if I want to keep at d10 or lower to d8/d6) that the druid of twilight did which for thematics was changed to force cause arcane energy and all that. I'm also still wondering what you guys think about the expended spell list I provided Just let me know what ya'll think about these changes.

You have made a pact with a mysterious archemage of ancient times. The pact that you made with the patron grants you a fraction of their knowledge while bolstering your spellcasting ability. They may have given you this pact so that you may gather magical artifacts, learn secrets of the arcane, or just as a pawn in their schemes. Some patrons that can fit this mold  could include any variety of Liches, Ancient Wizard’s, or even the infamous the momentary god, Karsus.

• Expanded Spell List
  • Spell level – 1st – Magic Missile, Identify
  • Spell level – 2nd – Levitate, Nystul’s Magic Aura
  • Spell level – 3rd – Fireball, Lightning Bolt
  • Spell level – 4th – Private Sanctum, Secret Chest
  • Spell level – 5th – Cone of Cold, Teleportation circle
• Level 1
  • Patron’s Teachings – At 1st level, you learn one wizard cantrip. They count as warlock cantrips for you, but they don’t count against your number of cantrips known. You also gain proficiency in the Arcana skill. Your proficiency bonus is doubled for any ability check you make with the Arcana skill.
  • Arcane Surge – At 1st level, your patron has granted you some of his arcane power to empower your spells. You have a pool of energy represented by a number of d10s equal to your warlock level. When you roll damage for a spell, you can increase that damage by spending dice from the pool. You can spend a number of dice equal to half your warlock level or less. Roll the spent dice and them to the damage as force damage. You regain all expended dice when you finish a long rest
• Level 6
  • Selected studies – Starting at 6th level, you learn two wizard spells of 3rd level or lower. Both wizard spells gained count as warlock spells to you.
• Level 10
  • Unrestrained Learning – Beginning at 10th level, you have become more ambitious in learning new spells. You learn one 5th level or lower spells from the wizard spell list, and you can cast this spell without expending a spell slot once per long rest. This spell counts as warlock spells to you.
• Level 14
  • Patron’s Secret – Starting at 14th level, you peered into the knowledge of your patron and discovered a great power. You gain 1 additional spell slot and once per day, you can cast a spell of your choice from the warlock or wizard spell list. The spell must be of level for which you can cast, or at the level of mystic arcanum known. If the spell isn’t a warlock spell, its counts as a warlock spell when you cast it. You regain use of this feature at the next dawn.

Arcane Surge is pretty powerful, a 10th level Warlock with magic missile would do 7d4+7+5d10 damage on one casting.

Quote from AaronWho >>

Arcane Surge is pretty powerful, a 10th level Warlock with magic missile would do 7d4+7+5d10 damage on one casting.

Would you think that lowering the damage die to d6s would be better or just change it to something like "for leveled spells you can add your CHA mod to a damage roll" ...personally I do like the idea of the pool of a resource, but I definitely see where you are coming from that; it was definitely an oversight. I just feel like the spellcasting mod bonus is over used, and would like to do something like a pooled resource or just something different. Another possibility could be giving advantage on damage rolls for a spell, with the ability to use it a number time equal to your CHA per long/short rest.

I also did see another oversight on the max amount of dice per use, should've been up to CHA mod rather than half warlock level, but that part wouldn't have fixed the issue that you brought up.