is there a set limit a day or a second that my app could send?
The API Rate Limit depends on the which type of token you use with your requests. If you make a request with just a client-id then you’re limited to 30/minute, if you use an App Access Token or a User Access Token then the rate limit is 800/minute.
Details can be found on the docs: https://dev.twitch.tv/docs/api/guide/#rate-limits
Some endpoints also have additional global rate limits, if these apply it will be mentioned in the reference section for that endpoint.
The deprecated v3 and v5 API don’t have a strict rate limit, but if you make excessive requests Twitch may take action, and the general guidance is to try not exceed 1 request a second.
What if someone had an app with a lot of people and hypothetically needed more requests in the future? Would you fill out the “Twitch Developer Limit Increase Request”?
If your needs exceed the default rate limit then yes you could request a rate limit increase https://dev.twitch.tv/limit-increase/
As long as you can justify why you need an increase in rate limit, and your requested limit increase is reasonable, then Twitch are usually happy to try accommodate your apps needs.
And if I get a user to link their account, its 800 request per minute for that user anytime they use my app to make a call on their behalf? last question i promise lol
As pointed out in the link that Dist provided, the Bearer (user) token provides 800 points per minute. This would be by user that authenticates with your application and you are granted a Bearer token to use on their behalf. Note “per minute, per user”