I think the title asks the question. There are key signatures with Ab, Bb, Cb, Db, Eb and Gb. But no Fb.
There's probably no practical reason for my question. I'm just curious.
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Technically, there could be, you just keep extending the pattern. You could even keep extending it to the point where you need to start using double flats, though this is almost never done in practice.
Though it is rare, wikipedia (linked above) points out that the key of C♭ has been used, and, in fact, is the most resonant key for the harp. The main reason that this key isn't used frequently is because it is enharmonically equivalent to the key of B, which only has 5 sharps instead of 7 flats, and is therefore easier for many instruments to play. Notice how each pair of notes from the following two scales are different names for the same pitch:
To expand, enharmonic equivalence is an invention of convenience. Musical intervals are just frequency ratios, and ratios with smaller numbers sound more consonant. For example, the octave is 2:1, and the perfect fifth is 3:2, the two simplest.
Compounding ratios by stacking intervals serves to create more notes. However, since no nonzero power of 3/2 can ever equal a power of 2 (prime factorization), that means you can always create new notes by adding on fifths and correcting the octave - the number of pitch classes is infinite.
To have something usable requires one of two things: restricting usage to some level of sharps and flats, or deliberately tuning the fifth wrong so that the cycle of fifths closes. Thus the art of temperament, or tuning theory.
results in Pythagorean tuning, where the keyboard would have (say) Eb, but D# literally does not exist. You cannot play the fifth G#-D#, and attempting to use G#-Eb, a diminished sixth, sounds horrible (also known as a wolf fifth, representing 192:125 or worse).
results in equal temperament (or 12EDO, equal divisions of the octave), where each fifth is slightly out of tune by about 1/50 of a semitone. In exchange, there are only 12 equivalent notes in an octave, and most intervals are tolerable. Combinations are possible, leading to meantone and other temperaments.
Twelve fifths is very close to seven octaves (hence 12EDO), off by the difference between B# and C, a tiny ratio known as the Pythagorean comma - which is discarded in Western music theory. Thus the origin of enharmonic equivalence: B# is represented by the same pitch as C, because we can't be bothered to care about the difference, not because they are musically equal.
In short, when modulating in tonal harmony, using equivalence is absolutely improper and summarily barred, whatever the savings in notation. To wit, modulating from Db major to Db minor would be I->i, but to go to C# minor is from I->bbii, a transition that nobody uses.
A quick example I can think of is Bach's Fugue in C# major, BWV842/2 (WTC I). For whatever reason he chose to use 7 sharps instead of Db, and so in bar 19 he writes a full scale in iii melodic: E#-Fx-G#-A#-B#-Cx-Dx-B#-E#. Yes, the imitable key of E# minor.
That said, since equivalence is used, there isn't too much point in starting a piece with more than 7 sharps or flats. Modulation extends the range up to about 10 sharps or flats, but that's basically the limit.