So, with Tasha's here, and the ability to swap ASIs and proficiencies around, the main reason to pick a race is flavor, as well as making use of their special racial powers like Goblin's Bonus Action to Hide and Goliath's ability to shrug off damage! So, let's hear y'all's best pitch for "I wanted to use this race with this class because they can do *this* and now I can!" combos!

I made a Yuan-ti Way of Mercy monk that turned out pretty cool. Not sure if it's an optimal combo, but yuan-ti is just very strong in general and the flavor is very synergetic with the whole plague doctor vibe of the subclass.

I'm sure I'll do a Gnomebarian at some point as well. But mostly I'm just excited to play Rangers.

Lizardfolk barbarians... goblin warlocks... orc warlocks... all of the solid fantasy tropes that didn't make mechanical sense before.

I have so MANY ideas flowing through my mind that it worries me a bit! Stacking racial modifiers with stat bonuses might be really powerful.

Imagine a Shifter with the new Form of the Beast Barbarian.

How about a Psi Warrior with their Protective Field coupled with the Abjuration Wizard's Arcane Ward. Lvl 5 Wizard at 20 Int = a 15 point Ward. The Psi Fighter would have 6D8 (at lvl 5) for an average of about 26 points of damage blocked. So at lvl 10 that's 40+ damage that can be ignored or prevented.

The only thing I've seen that MIGHT be broken is a lvl 2 Barb/Lvl 6 Artificer/Armorer. With Half-Plate, infusions and a shield you could have 20 AC, Resistance to weapon damage, Thunder Gauntlets to punch with plus other infusions and gear. Lvl 20 AND Resistance seems WAY tanky to me!

Yuan-ti Circle of the Shepherd Druid that uses Conjure Animals every combat and always summons Constrictor Snakes, Giant Poisonous Snakes, or a Giant Constrictor Snake.

Meh

Goblin Lore Bard Archaeologist who uses his history and lore skills as well as his bard abilities to uncover every little secret and bit of lore to find out every known and quasi-known bit of information out there. 

Then sells it to the highest bidder.

No need to fight or even summon. Stop thinking small! ;)

I'm thinking about a kenku psi knight/great old one warlock. They use the power of the mind to perform magic and other things. Also, orc druid/warlock, aka a super shaman. I'm thinking fathomless warlock, with stars druid? That's probably my favourite combo. 

Half orc caster? I mean that's just scratching the surface, there is a lot of possibilities that could be amazing. I'm sure I have more, I will post more later.

Yuan Ti Pureblood Cavalier Fighter who rides a Stench Kow.

Goliath rogue. An 8' tall dude hiding behind a potted plant. Wielding a shortsword that looks like a butterknife in his giant hands.

Quote from HailRobonia >>

Goliath rogue. An 8' tall dude hiding behind a potted plant. Wielding a shortsword that looks like a butterknife in his giant hands.

 
My wife has been playing a female Goliath rogue for months. It's hilarious.

Yuan-ti Beast Barbarian who reveres the Primal Spirit of the World Serpent.

Aarakocra UA Phoenix Sorcerer, the ultimate fire bird.

Kenku Swarmkeeper Ranger, mob boss and his gang of murderous crows.

Minotaur Eloquence Bard, southern gentleman who plays the cowbells.

Simic Hybrid Fathomless Warlock, for when you really want to be a sea monster so badly.

Variant Human Hexblade Warlock with the Aberrant Dragonmark and Shadow-Touched Feat, ancient Shadow Dragon is his mother or father as well as his patron.

Grung Path of the Beast Barbarian, a wolverine frog

Kobold Rune Knight Fighter with a Halberd

Goliath Armorer Artificer (Guardian Type) 

Mountain Dwarf Bladesinger

Shifter Hexblade Warlock

A poor choice would be a Changeling who swaps Charisma and Deception proficiency for literally anything else.

Just becomes some wierdo wandering around town doing bad impressions.

oh dear, sweet child...

You now pick a race for it's mechanical benefits.  Mountain Dwarf Casters, Yuan-ti or Satyr barbarians or Drow anythings with two scimitars.  honorable mention to half-elf anythings that don't need armor/weapons from Mountain Dwarf.  Or variant human to get a tasty feat at first level and unlock darkness/devil's sight without MC, or things like fully powered abjuration ward wizards via armor of shadows invocation without MC or svirfneblin + racial feat.

Oh man, I thought a Gnome Barbarian was just a joke. Forgot about Gnome Cunning. Giving a Barbarian advantage on all mental saving throws against magic is pretty formidable.

Protector Aasimar Moon Druid would allow for a Flying Dire Wolf, or a Flying Giant Octopus at 4th level. (Grapple, fly, drop.)

Firbolg Rogue would get to turn invisible as a bonus action, while also getting Detect Magic, and Disguise Self for free.

Hobgoblin Rogue gets a bonus to attacks, saving throws, and skill checks if they fail, which is great for disarming a really dangerous trap, or getting Sneak Attack when you really need it.

Lizardfolk ??? would be able to reallocate their ability score to Dexterity to synergize with their Natural Armor.

Yuan-ti Sorcerer with Immunity to Poison and Transmuted Metamagic, you can drop Poison Fireballs at point blank range, or hide inside of a Poison Wall of Fire. (This is more about the new metamagic, rather than Character Creation options)

Tortle Wizard/Sorcerer/Warlock for the free AC 17. Shell Defense is a bit of a waste, but you could hold concentration while in your shell.

Shifter (Wildhunt) Barbarian Combine Reckless attack with Shifting Feature, "no creature within 30 feet of you can make an attack roll with advantage against you..."

Quote from Memnosyne >>

Shifter (Wildhunt) Barbarian Combine Reckless attack with Shifting Feature, "no creature within 30 feet of you can make an attack roll with advantage against you..."

 
Throw in a Rogue multiclass and a finesse weapon for a Sneak Attack per turn for 10 turns (9 can be +rage damage too)

Mountain dwarf clerics that don’t need to put 15 into STR to actually wear plate.

That's not really a "wacky" combo, but it certainly is mechanically sound. Especially with the rules from Tasha's.