Here are some cautionary remarks, plus references. You ask: Is there a more comprehensive list of such polynomials? The answer seems to be no. Lists get long very quickly, and as I commented earlier there is a built-in labeling problem: how to label each group element uniquely while working systematically with pairs of elements related by the Bruhat ordering?
If you focus especially on symmetric groups (or other finite Coxeter groups), the computational problem for each fixed group is a finite one. But already for $E_8$ the Lie group project cited by Paul Garrett has involved a huge effort to compute even the more limited list of Kazhdan-Lusztig-Vogan polynomials relevant to the study of unitary representations of a real Lie group. Here as elsewhere, computations are best done in a motivated framework where supporting theory exists to point toward likely uses for the information encoded in the polynomials.
For symmetric groups, there is the cautinary result of Patrick Polo, showing that every polynomial with non-negative integral coefficients and constant term 1 arises as a Kazhdan-Lusztig poluynomial for some pair of permutations related by the Bruhat ordering. This was announced in a bilingual Comptes Rendus note (1999) and explained in more detail in English in the online AMS journal Representation Theory here.
It's also worthwhile to look at Soergel's alternative development of the polynomials, avoiding mention of the $R$-polynomials (which haven't so far had a useful homological interpretation of their own): see his article in the same journal here. But his work, as in earlier cases involving algebraic geometry, combinatorics, representation theory, hasn't relied on first compiling lists of the polynomials.