There is no downside to nested encryption, except of course the performance overhead. But this only really makes sense if each layer has an independent key and each layer uses an algorithm from a different family. Improper key reuse weakens the scheme.

For symmetric cryptography like AES the benefit is dubious. It is far more likely that the content is decrypted because the key was acquired independently than that AES would be broken.

However, there absolutely is a benefit for asymmetric crypto and key agreement schemes. This is how current Post-Quantum Cryptography schemes work, because:

• commonly used algorithm families like RSA and Elliptic-Cuve will be broken as soon as a sufficiently large quantum computer exist
• proposed PQC algorithms are comparatively immature, and some of them will be broken in the near future

Nesting one algorithm from each family gives us the best of both worlds, at a performance overhead: conventional asymmetric cryptography give us temporary security in the near future, and the second PQC layer gives us a chance at long-term security.