short answer: less commonly, C flat and C sharp major, and A flat minor with 7 accidentals each. more commonly, F sharp/G flat major, and E flat/ D sharp minor, each with 6 accidentals.
accidentals are also used in the key signature to define the key.
long answer: the most accidentals will be in C flat major or C sharp major, 7, but those keys usually are spelled as B major and D flat major, respectively. however, there are instances of both. the minor equivalent A flat minor also occurs, and is more ocnvinient than the usual G sharp minor spelling, because in ascending melodic minor G and F natural (raised 7th and 6th) are more convinient than F and E double-sharp.
more commonly the keys with most sharps and flats are G flat/F sharp major, or E flat/D sharp minor.
in the middle of pieces (but without a key signature, just what becomes implied by the accidentals) you can find F flat major (6 flats and one double flat), and G sharp major (6 sharps and one double sharp), D sharp major (rarely, with 5 sharps and two double sharps) as well as A sharp minor (in the D sharp minor prelude and fugue - book I by JS Bach, it has 7 sharps, but in the ascending melodic form 2 of them become double sharps - nightmare), D flat minor (6 flats and one double flat, but not so bad in the ascending melodic form).