Ohhhh my favorite topic! (AKA the only one I know something about)
To go back to the OP's question, producers often set official 'minimums', that are typically $10k for a broadway play and $25k for a musical. Sometimes, if something is expected to be a bit hit, you'll be capped at what you can invest too, based on your relationship with the producer / how early you sign on. I had heard that board members of The Public were generally 'allowed' to invest no more than $25-50k each in Hamilton, where as a show like Gentleman's Guide would have been happy to accept a few million
Now, this is if you invest directly with the producers (aka are signing the subscription documents). My very first investment was only $2,500 by using a special purpose partnership (lets call it in this case I Want To Invest In Shows series A, LLC), which bundled investments until it had $25,000, and then that LLC became an investor in the show. This is more common than you might think, as people even with bigger checks often do this to get higher perks.
As far as negotiating goes; most shows have a 'favored nation clause' that prevents an investors from individually negotiating, because the producer would then have to go back and offer every other investors terms at least as good as every successive investor. That said, there is room for some negotiating with what share of a producer's cut an individual investor or group of investors can receive if they come in with a big enough check (talking $1mm+ here). The things they negotiate might be getting their name above the title, receiving 'off the top' profits, torchbearing points for providing front money, etc. But there is no negotiating as far as "I want 10% of the show for my investment".
Lastly, as an anecdote, I know many investors and co producers who simply do it to go to the party and take some pictures and be in the room where it happens. That said, I've invested in 5 shows and each of them have recouped. Not that I'm any sort of sharp eye, I think I've mostly gotten lucky. But all of the people I invest with are also looking to make money :)