The main benefit is that all the ciphers need to be broken to compromise the data. So as long as each of the ciphers are strong, you'll get a little better security.

The main downside is additional computational complexity. Most disk encryption systems have minimal performance impact because the CPU has hardware acceleration for popular ciphers, so it can keep up with disk reads. Both of my computers use full disk encryption, and the system runs plenty fast. If you run multiple ciphers, you're more likely to notice the decryption process.

The likelihood of any strong cipher being broken is incredibly low, and it's much more likely that an attacker will compromise the data while it's unencrypted than attacking the cipher directly, in which case the cascading cipher won't help.

If you want paranoid levels of security, consider following the NSA's Rule of Two, which means two completely independent layers of encryption. Don't use two ciphers from the same vendor, but two vendors. For example, use full disk encryption through the OS, and an encryption application for important files. If you use two ciphers from the same vendor (i.e. your application with cascading ciphers), it's more likely that they would share a vulnerability than if they came from different vendors.

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.