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In music theory when naming intervals, a lowered diminished interval is a doubly diminished interval (dd) and a raised augmented interval is a doubly augmented interval (AA). Ex a C# to a Gb would be a doubly diminished 5th (dd5) and a Cb to a F# would be a doubly augmented 4th (AA4). However, these intervals can be converted to their enharmonic equivalent so the make more sense so instead of having a C# to a Gb you could write it as C# to F# and have it be a P4 and instead of having a Cb to F# you could write a B to F# and have it be a P5.

I have never actually seen a naturally occurring doubly diminished or doubly augmented interval I have only isolated examples in theory books. Is there ever any scenario where any doubly diminished or doubly augmented intervals are used in melodies or chords?

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The last piece in Ligeti's 'Musica Ricercata' is a good example of double diminished everything. But these intervals in the piece occur only because of poyphony. I'm quite sure that if you dig up some early or very late Shostakovich you'll be able to find examples of more harmonic and melodic (rather than polyphonic) use of double diminished intervals. Shostakovich is a very frequent user of diminished and double diminished intervals.

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Excellent, there they are! Including the inversion of the exact same interval I hypothesized about in my answer, Gb–C# doubly-augmented fourth implying resolution to F–D. Thanks so much for providing an example, I'm going to show it to my students next week! –  Pat Muchmore Feb 23 at 14:20
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Besides for user181381's excellent example, I was looking around last week and found that altered chords may have a doubly augmented or doubly diminished interval in them.

Let's look at a C7b5#9 and a C7#5b9

C7b5#9 - C  E  Gb  Bb  D#

C7#5b9 - C  E  G#  Bb  Db 

In the case of a C7b5#9, the interval between Gb and D# is a doubly augmented 5th and the Gb wants to go to an F(tonic) and the D# wants to go to an E(tonic's 7th aka leading tone). In the case of a C7#5b9, the interval between G# and Db is a doubly diminished 5th and the G# wants to go to an A(the 3rd of tonic) and the Db wants to go to a C(the 5th of tonic).

So these intervals just help lead back to tonic and function as an extension of a dominant chord.

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In standard functional harmony, diminished intervals only naturally occur between the 4th and 7th scale degrees, which is typically found in a vii° or V7 harmony. Double-diminished or double-augmented intervals don't occur anywhere in this system in its most basic version. Much of our Western music is based on this harmonic system, so intervals of this type are naturally much less common.

But the more inventive and expansive harmony becomes, and the more chromatic a piece of music is, the more likely you are to find intervals of this sort. Usually they arise out of linear motion than a vertical, simultaneous chord with a functional harmonic structure, though.

I recall seeing lots of chromaticism of this type (leading to the intervals you seek) when I was analyzing the music of Scriabin and Berg in school. That's a good place to start.

Unfortunately I can't think of any functional harmonic chord structure or progression that contains a doubly-diminished or doubly-augmented interval.

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I haven't seen them in an actual piece, but I'm hoping somebody else on here has, because I'd be very interested. However, I can think of a functional reason to use your example, C#-Gb; it would make sense as a chromatic dissonance that resolves to a D-F minor third. Doubly diminished sevenths such as C#-Bbb would imply "resolution" to a o5, in this case D-Ab. They should absolutely never be used unless there's a specific implied resolution like this, and there's a pretty strong case to be made that they shouldn't be used even then because they're just too unfamiliar to a performer. Personally, I think there's something to be said for truly showing the actual function despite odd intervals (for example, I often use augmented 3rds in my own music, implying an outward resolution to a P5), but I just don't know if I could bring myself to throw a doubly diminished or augmented interval at somebody.

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I've seen one somewhere, but can't remember where. Probably in some "innovative" 20th century music somewhere. –  BobRodes Feb 21 at 15:12
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