How exotic is too exotic for a player race?
That is a completely subjective, table-by-table, even person-by-person question. It has no actual answer.
I have been pondering a setting with just a ludicrous amount of racial diversity where you would have telepathic goldfish doing business with demons and drakes. Just any crazy/fun race idea gets thrown in. The question is would it be possible to have a PC party where each member is radically a different race.
It is certainly possible. How advisable "just any crazy/fun race idea" would be is, again, completely subjective.
I'm sure it can. It would depend, obviously, on what these creatures/characters are...and how much 'balance" are you talking about? Just mechanically balanced? Sure. 5e can do that. "Immersion" balanced/making sense? That's a separate question. But can certainly be done. As others have said, it just requires more work on the part of the DM and mthe willingness of the players. Someone's idea of a mouse fighter with a 20 strength? Assuming the anthro-intelligent-mouse character was a go, the 20 strength, to me, would be a no way in the 9 Hells.
"Fun" is a subjective, individual -and not even consistent within a single individual- word. So, yeah. Sure it can be...if you think it will be and want to have "fun" with it. Then you certainly can.
As someone else mentioned, a lot of the "fun" of these kinds of "freaky/weirdo/unique/exotic/special snowflake" characters are that they ARE the "odd man out" among a party of "normal" characters in a largely "normal" [to the parameters of D&D] fantasy world. A party of entire city/nation/world of them does not suit everyone for a "fun" time.
My issue is immersion and suspension of disbelief [which is just as subjective and individual as "fun"]. Can I, as a human mage, have fun with a telepathic goldfish in the group? Probably. Can I have fun as an elf ranger in a party FULL of telepathic goldfish? A world of telekinetic rocks and sentient gas clouds? A lot less likely.
When I play D&D, there are certain expectations and [I prefer] assumptions about a given setting and the proposed genre: low through high fantasy, the presence of magic, relatively rare or low tech, etc... whatever the assumptions of the game's proposed setting and genre are.
I don't want to walk into a game of D&D and be faced with a "supers" game or a game of Toon...or, at the other side of the spectrum, historical accuracy, wild west or high tech sci-fi. Of course, I wouldn't be walking in in the first place since, as a player, I would/should know what those assumptions are before play begins.
If your players are on board with this, knock yourselves out and have a blast.
How much diversity is too much?
There is no answer here. It's all just personal preference and individual enjoyment.
Just, for my own tastes, it seems with the bulk of D&D settings and OSR type fantasy-focused games, you start topping out around 10 to a dozen. More than that begins to "jump the shark", as it were, or muddies the waters of the setting just too much for my enjoyment. And if you have a whole party of "exotic", the DM after a while just can't be bothered with having every village freak at your arrival...or you're in a a setting where everyone is special...so then, what's the point? When anyone can be/do anything...no one is actually "exotic" because exotic is the norm.
Then as a fun aside, feel free to post your most out there race idea.
I don't know if I can go more "out there" than telepathic goldfish and psionic rocks and clouds...as a race. Once you're out of the "resembling anything humanoid"...or even "biological", I'm not sure there's anything "more out there" to add. A character of sentient energy that only exists when it wants to [which is really no different or more creative than a sentient gas cloud or rock]? How about we all just play omnipresent omnipotence, then? Game's over. We won.