There is a refinement of the connect-sum decomposition called 'satellite operations' or 'splicing'. Like the connect-sum operation, it's originally due to Schubert.
Unfortunately, this is infinitely-many operations. You could think of it as one operation that takes as input $n$ knots and an $(n+1)$-component link $L = (L_0, L_1, \cdots, L_n)$ such that the sublink $(L_1, \cdots, L_n)$ is trivial. You also demand that the complement of $L$ in $S^3$ is either Seifert-fibred or hyperbolic. It's also an essentially unique decomposition (like the connect-sum decomposition) -- the link is well defined but the indexing of the knots is only unique up to the symmetry group of the link $L$.
<img src="http://upload.wikimedia.org/wikipedia/en/a/ad/B_sat2.png"
<img src="http://upload.wikimedia.org/wikipedia/en/7/7c/Knot_with_borromean_rings_in_jsj_decomp.png"