Any disadvantages of using multiple encryption algorithms on same data?

One of the stories at the beginning of Volume 2, of The Art of Computer Programming tells of Knuth's early attempt to generate random numbers.

It's a story of a clever kid, relying on intuitions and getting it wrong, because it's mathematics.

Thus he devotes one of his twelve chapters entirely to random numbers.

Unfortunately, it is well established by internet forums that nobody should read Knuth.

I agree. There are various complexity downsides (implementation, key management, etc) but it should be fine from a practical cryptographic security perspective.

While I can't offer you a formal cryptanalysis, I do have a handwave-y thought experiment & proof by contradiction for intuition:

Consider two encryption algorithms A[...] and B[...] that are "good" in the standard ways (adversary cannot decrypt messages without keys, keys cannot be leaked by chosen plaintext attack, etc).

If putting A inside of B like B[A[...]] renders the system insecure, it would imply that B[...] was not a "good" encryption function! (if it was good, it would not break just because the input message happens to be the output of A[...]).

Therefore by handwave-y proof by contradiction, if A[...] and B[...] are "good", then A[B[...]] should be OK.

(Disclaimer: Again, this is a casual thought experiment and not real cryptanalysis. This result may not be true under formal cryptanalysis.)