Having more than two prime factors is already supported by the PKCS#1 standard. This is called a "multiprime RSA" algorithm.

On the plus size, this may offer some computational performance improvement via the Chinese Remainder Theorem. For instance, if you use a modulus with $k$ factors, the CRT speedup factor is about $k^2.$

However, using too small factors may weaken the modulus. The best known factorization algorithms depend only on the modulus unless the factors are small enough to enter the range feasible with Elliptic Curve Multiplication which has a cost which depends (mostly) on the size of the smallest factor.

More generally, a batch RSA algorithm (not multiprime RSA) can be used to speed up batch processing of many RSA signatures at once in a server setting, with substantial speedups. The paper by Boneh and Scacham here describes these ideas.