Is there an asymmetric algorithm that can perform double encryption?
No. ​ For semantic security, the encryption of a known plaintext with a given key must be overwhelmingly likely to have high entropy so for most plaintexts and keys, there must be lots of possible encryptions of the plaintext with the key.

So for most keys, the ciphertext space must be significantly larger than the plaintext space.



You might be interested in trapdoor permutations. The only candidates I'm aware of for [such permutations with domain not depending on the public key]
come from multi-variate crypto.

On the other hand, it's easy for symmetric schemes to satisfy the deterministic version of CCA-security.