: What is the interval from C double flat to E double sharp called? What is this interval called in terms of relative scale degrees?

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It would be a quadruply augmented 3rd and would be written as AAAA3.

C to E alone is obviously a major third (M3). From there realize that by adding a double sharp to the E you increase the distance by two semitones the and the resulting interval C to E♯♯ is a doubly augmented third (AA3). From there you can see by lowing the C two semitones results in the interval being increased by two more semitones resulting in the quadruply augmented 3rd.

Doubly augmented intervals in general are rare as seen in this question, so I doubt you'll see it outside a theoretical context.

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When you come across something like this, you should remove all the accidentals and add them one at a time:

C- E: major 3rd
C- E#:augmented 3rd
C -Ex: double augmented 3rd
Cb- Ex: triple augmented 3rd
Cbb-Ex: quadruple augmented 3rd.

Which I doubt you'll ever meet, even if it exists. Most likely it'll be Bb -F# which is a augmented 5th interval or Bb - Gb which is a minor sixth.