Dragons, dragonborns (though half-dragons are original and better than dragonborn but more poweful so they wasn’t balanced as player race) and kobolds.
Khajiit and all other cat-people and cat-monsters everywhere (maybe except Kzinti)
Owlbears are great too.
Trolls come in dizzying array of variants (my favourite are the troll gods from Edding’s books).
Tengu, especially the crow-tengu

Do biblically accurate angels count?

I love Goblins as well so I always make mine Pathfinder inspired so they will be green with a big head who love fire and general chaos instead of orange like dnd goblins. They are always fun since they like to straddle the lines between disgusting, cute, destructive and helpless. I know my players always will befriend them so I like to put a lot of them in there.

I also like to treat them like cockroaches or fruit flies who breed very fast and can adopt to any conditions. So there will be strange variations based on where they live. Mud goblins, fire goblins, moss goblins who just have small physical adaptations to better fit their habitat.

I don’t know if you would call Hobbits Halflings monster races but they are my favorite. I love the way Eberron did them with their mark of hospitability so they have special magic to make everywhere more comfortable and good things just happen to them. I love how they just took the fearlessness and everything good from Kinder and just wrapped it into regular hobbits halfings.

To play as in RPGs, I like big stuff and little stuff. Like orcs and goblins. Or very non humanoid stuff like slimes.

In general, idk, I used to really love gnomes (warcraft/d&d style).

Edit: I totally missed the word monster in the title. I like shapeshifters, oozes, any sort of undead besides the typical zombie and ghost, and probably most of all are demons.

Sphinxes or lamia. The cruel intelligence of man with the refined predatory abilities of felids/snakes. What a match!

I feel like Beholders are the product of some nightmare fueled fever dream. They fascinate me endlessly.

Weeping Angels

i love lizardmen. mainly because i am a scalie.

Probably dwarves - they’re not that exotic but I really vibe with them… for something more out there I’m a big fan of Yuan-Ti, they have spectacular lore and it’s always tickled me that their most human-like form is basically considered low-born while the pure bloods are full on snakes.

Dwarves definitely take the cake for me though, big beards, stout, egalitarian, sometimes greedy - but always devout craftsfolk. As a big gender non-conforming man with a bigger beard and an intense love of my craftwork, I really vibe with them.

… I am a dwarf, and I’m digging a hole Diggy diggy hole, diggy diggy hole

Kobolds deserved the place in the player’s handbook that dragonborn got. Those little scrappy fuckers maybe being the actual scions of dragons appeals to me in a way that dragonborn just do not.

I know it makes me super basic but… dragons. I know, it’s not inspiring. But I must add a caveat. I prefer that they are intelligent, on par with or surpassing humans in intelligence and willing (if reluctantly) to interact with them. Game of Thrones dragons are cool and all but they don’t really do it for me in the same way as, say, the dragon from Dragonheart.

Pushing the boundaries of the question, but 40k orks! How can you not love the big green lugs?

I’ll go with the Alzabo from Gene Wolf’s worlds. Mimics the intelligence of anything it eats and begs its prey to be eaten in the voice of (already eaten) loved ones.

The people who submit, don’t do so out of momentary stupidity, but because the Alzebo/Loved Ones make such a compelling case that the only way to be reunited again is to join the beast.

Tieflings. The “alignment” section of the 5e PHB (before they decided describing alignments was racist and removed it) read:

Tieflings might not have an innate tendency toward evil, but many of them end up there.

Which is such a powerful storytelling device. It does what sci-fi and fantasy are so often great at: comment on real-world social issues with a step of indirection that makes the story feel less on the nose. Their internal innate selves are indistinguishable from humans, but because they have horns, a devil’s tail, and often reddish skin, people assume they’re evil and treat them accordingly.

It’s an element that is handled so excellently by Erin M. Evans in her Brimstone Angels series:

A woman stood in the doorway opposite the bench, watching Farideh with a wary eye, no subtlety in her distaste. Farideh shifted uncomfortably.

“You waiting for someone?” the woman said after an interminable time.

“My friend,” Farideh said. “He won’t be long.”

“Buying spices from another devilborn.” She sniffed. “Your kind do like to stick together.”

Farideh’s tail flicked nervously. She pulled it closer to lie along her thigh. “My friend’s human, many thanks.”

“Is he now?” Farideh met the woman’s skeptical gaze. Without the ring of white humans were used to, Farideh’s eyes were unreadable. Emotionless. Inhuman. The shopkeeper could stare as long as she liked and Farideh knew she wouldn’t see anything there, not without practice.

“Do you want me to have him show you?” Farideh said. “Or do you want to say what it is you’re getting at?”

Farideh knew perfectly well what the shopkeeper was getting at: she didn’t belong here. Whatever clientele the shopkeeper was used to dealing with, a seventeen-year-old tiefling trying to rein in the tendrils of shadow that curled and coiled around the edges of her frame was not a part of it

Longer excerpt available on author’s blog. (It’s book 3 of the series, but no significant spoilers here.)

Of course that’s only one small part of the characters, but it’s done so well. They’re well-rounded full people who, like any real human, have to deal with getting through life (in their case, fantasy action adventures) while other people react to them.