A double-sharp would raise a flat note by a tone and a half (three semitones).
A double-flat would lower a sharp note by a tone and a half (three semitones).
Having said both these things, if you have a piece that is changing a flat note to a double-sharp, or changing a sharp note to a double flat, it is likely that you could use a much better enharmonic spelling. For example, it seems far more likely that you would move from G# to E# (or F, depending upon context), than G# to Gbb, even if you wanted to use, for instance, Fb, Eb and D close to these notes too. In fact, if music is very chromatic, it often makes sense to use fewer double-sharps and double-flats, even if this requires using a few more accidentals (naturals, for instance). This is because very chromatic music can start to lose its connection with major or minor scale degrees, this chromatic "alteration" of scale degrees being a good reason to use double-sharps and double-flats in the first place (either as chromatic alteration within a particular key, or chromatic alteration which produces a modulation to another key). In the example above, it would make sense to use the "simplest" accidentals (Ab F E Eb D, G# F E Eb D, Ab F E D# D or G# F E D# D), to make it easy to read. Which of these would be used, would depend upon whether these are heard within a diatonic context or not, and if so within which key. (If, of course, these are used during a passage in F# Minor, it's likely that an E# would be used - context is everything.)
I suppose you could also adjust a double-sharp note to a double-flat (and vice versa), in theory, which would be a a movement down of two tones (four semitones). I can't think of a reason why you would want to do this, apart from as a theoretical exercise, though! If these two notes were next to each other, I guess this would be the interval of a quadruply augmented unison!! (Thanks for correction, @PatMuchmore.)
EDIT: as @Guidot points out in his answer, this situation is basically theoretical; it is very unlikely to actually appear in practice. Double-sharps are nearly always used to extend a sharp by a further semitone (usually where the sharp already appears in the key signature); double-flats are nearly always used to extend a flat by a further semitone (usually where the flat already appears in the key signature). For this reason, in practice it is very unlikely that there would be a need to move from flat to double-sharp, or sharp to double-flat. (As I point out above, if one ever finds the need to do either of these things in a piece of music, the enharmonic spelling is almost certainly incorrect.)
Also, @CarlWitthoft has asked me to be clear: any accidental (sharp, flat, natural, double-sharp, double-flat) "cancels" the associated key-signature sharp or flat for the same note. So, in theory, this could be a way that a note gets "changed" by three semitones by using a double-sharp or double-flat. An example would be using an A double-sharp where the key-signature has an Ab (Eb Major for example). But, again, this is extremely unlikely to be necessary, for three reasons:
- an A double-sharp is only really needed if you were also using B# and C# in a passage, too; this is very unlikely in a key that has an Ab in the key-signature (C and Db would be used).
- as per @Guidot's answer, an A double-sharp is most likely to be used to "extend" an A# (from the key signature of C# Major, for instance, where there is also a B#).
- if the music is very chromatic or atonal, it would be better not to use a key-signature, in which case it is usually appropriate to use accidentals which are easiest to read, so any Ab used in a bar would move to a B natural, rather than an A double-sharp.