For a broad reference, see the Airplane Owners and Pilots Association (AOPA) database on health conditions that may affect medical certification.
https://www.aopa.org/go-fly/medical-resources/health-conditi...
But if you're not a paid pilot, you're better off not going down that route, since once you're on record as having a deficiency, that's a permanent concern to the FAA. And if they decline the special issuance, you're also not eligible for the drivers license medical.
I would advise nobody with issues who wants to fly for hobby reasons to go down the PPL route since you can get banned for failng a medical. There's almost no upside and huge downside to risking that.
I did witness an AME re-approve a commercial guy with a DUI though while in the waiting room. Good on you, FAA.
Thanks for the DUI anecdote. Great that he got re-approved, but that can be retracted anytime.
Just to recap - the FAA assumes if you get an airman medical, you might try to carry passengers, so their medical rules are maximally strict.
The 2 better options are:
1) Do some research on Sport Pilot and LSA planes. No FAA medical or PPL required, just a valid drivers license.
2) Get involved in the ultralight community, ask around about training, and buy a Hummel. No FAA medical or license required, but I'd recommend doing PPL-level training for safety reasons.
Just for those reading along: all FAA licenses require demonstrating mastery of the aircraft to certain tolerances. PPL has pretty basic requirements that get you from Point A to Point B, but no acro or IFR, so it doesn't make you an ace. But a PPL is still a world away from an untrained ultralight pilot.