I'm trying to benchmark some NMR chemical shifts. The literature suggests that the pcS-4 basis set will be the most accurate. I was wondering if there were any basis sets that are more complete than the Jensen basis sets? Even for just general basis sets. I'm looking for something similar to the Partridge basis sets, except for DFT.
Using pcS-4 only tells you how much the error from your basis set is. This doesn't necessarily have any link to the total error, which comes from basis set, DFT functional, molecular geometries, treatment of solvation, and various other aspects of your chosen method. So, if all you want to do is to investigate the effect of basis set while keeping everything else the same, then sure, go ahead and use the pcS-n basis sets. They were designed with convergence to the complete basis set limit in mind, so pcS-4 is probably as close to the CBS limit as you can reasonably get. For a recent (and very thorough) example of benchmarking using pcS-4 (technically the segmented version pcSseg-4), see J. Chem. Theory Comput. 2018, 14 (2), 619–637.
If your aim is just to get the computed shifts as accurate as possible, then CCSD(T) with large pcS-n or cc-pVnZ (or even extrapolation to the CBS limit) is probably what you want. For benchmarking you're not likely to be using any excessively big molecules anyway, so for a one-off calculation I imagine the computer time needed is quite tolerable.
As far as DFT is concerned, as has already been pointed out, if you don't use a good functional then no matter what basis set you use, the shifts will not be great anyway. Hybrid functionals like PBE0 or mPW1PW91 are popular in the literature (and B3LYP of course), but there are some other interesting ones like M06-L (see J. Phys. Chem. A 2008, 112 (30), 6794–6799), and recently double-hybrid functionals have been suggested (J. Chem. Theory Comput. 2018, 14 (9), 4756–4771).
Aside from that, there are also empirical ways of increasing the accuracy of DFT-predicted shifts, which you may or may not want to take into account when benchmarking. A popular one is linear scaling of shifts (the slope and intercept being determined from a given test set). Tantillo has a nice collection of these scaling factors at http://cheshirenmr.info/index.htm. Although some of the work cited can be a bit outdated in such a rapidly-moving field, the methodology itself is still highly relevant.
First and foremost: there is no best or most complete finite basisset (except for trivial examples). The only complete basis is your infinite dimensional Hilbert space.
You may phrase it better like this: With the current state of computer hardware and in combination with your ab initio method of choice applied to a specific model system this basis set gives results with an acceptable error for your variable of interest and is still computationally feasible.
One good test is always to increase the basis set. If your variable of interest is not affected, you are probably converged with respect to your basis set.
Note that it really depends on your variable of interest. If you want to find equilibrium geometries then hartree fock + smallish basis set is often enough. For reliable chemical shifts you need to push the basis set and method really hard.
The comment suggested wavefunction based methods, because they can be consistenly improved by increasing the number of excitations. If you can use only DFT, you should check at least two different functionals to see if they agree.