Every cipher I've encountered seems to be built from slight variations of only three basic operations: the S-box, the P-box, and the XOR. I’m just wondering, are there any others? And if there aren't (keep in mind I'm not counting XNOR here, since it's semantically the same as XOR), is there any way to prove that those three operations are the only ones that have the desired entropy properties needed for cryptography?
Actually the list is large. You're only describing the old stalwarts that have been around for donkey's years in say DES. Substitution and permutation form what is termed confusion and diffusion. These aren't always required as there are alternatives. Your sentence regarding entropy as a function is inaccurate as in cryptography the entropy is external to the primitive, eg. in a key or password.
If you consider all primitives and not just ciphers, Keccak hash uses a quite elegant permutation of bits in it's $\pi$ permutation which isn't like a traditional P box as it does not distribute active bits between multiple S boxes. Whirlpool hash uses Galois field multiplication to achieve diffusion, as does Rijndael /AES.
Stream ciphers operate differently, and a stream of pseudo random numbers are used. ISAAC and RC4 are examples that use various array transpositions /indirections and bitwise rotations. I guess if you had a lot of time, Blum Blum Shub could be seeded to product a (slow) cipher stream of pseudo random numbers that is based upon prime multiplication and quadratic residues.
If you then expand the scope to asymmetric techniques, that are all manner of other mathematically hard trap door problems that can be used in things like lattice and elliptic curve cryptography, key exchange etc.
There are even alternatives to XOR, such as modular addition used in Speck. Speck also extensively features the bit rotate operator.
So you see that the list is longer than you thought, and I've only included a few of the operations that underpin cryptography. I hope that by listing these, it's proved by deduction that S and P boxes and XOR are not the only tools.