I can't recall a time when everything I needed fit in one uke case!
What leaves the house with me when I meet up to play with others: two ukes (reentrant and low G), each with a strap and a clip-on mini tuner, each in a separate hardshell case; extra strings for both; one giant notebook (it weighs 10 pounds!) of sheet music; one heavy duty folding music stand (see aforementioned 10 pound notebook); and my usual non-uke stuff that includes four pairs of glasses (bifocals, sun, distance, and middle-distance), wallet, keys, cellphone, pouch of useful things like nail clippers, mini Leatherman, and so forth, notebook and pens.
What doesn't leave the house with me: another 10 pound notebook of chord charts dating back to my first ever uke class many years ago.
What inevitably happens when I meet up with others to play: someone always, ALWAYS want to play something that we haven't looked at in years, meaning it's in that notebook that didn't leave the house with me.
There are only two possible solutions to this last problem: learn to play by ear, or give in to the call of the iPad. Since I refuse to do the latter - guess I'd better get on with the former!
ETA: I should clarify that these days when I leave the house to play uke, it's usually in an instrumental trio or in an instrumental ensemble, so my needs might not be typical. Back when I used to take a beater uke to the beach or park to play alone, I just took the uke - didn't even bother with a case.