Would it be possible to establish a TLS connection over TLS with OpenSSL or some other tool?
If possible, would the certificates for each level need to be different?
This should work just fine in theory, though I cannot say for sure whether OpenSSL or something would support it easily. You can technically use the same certificate for multiple TLS connections, even if one is nested inside another.
However, I want to point out that one common reason to nest TLS connections might be to tunnel data over a multi-layered encrypted connection, making some subset of the data available at each stop in the tunnel (i.e. peeling back a layer of the encryption). Using the same certificate doesn't really support that use case. Perhaps you've got another use case in mind.
Furthermore, it is cryptographically sound to encrypt encrypted data. That is, more encryption cannot make data less secure. Lastly, encrypting encrypted data alone will not make it more secure. That is, AES(AES(x,key1),key2) where key1 != key2 is not more (or less) secure than AES(x, key1). Just in case that was your motivation.