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.