Which language has the largest number of vowels with minimal pairs?
This is one of those "it depends" questions. Dinka (Bor dialect) has the vowels [i e ɛ ɔ o u a], as well as long and over-long versions of these (21 vowels), and 4 phonatory contrasts (breathy, hollow, model, creaky) → 84 vowel, which can have 4 different tones (H, F, L, R) giving 336. Unfortunately they do not also have nasal vowels.
You could redefine the question to as to eliminate properties like tone or length that can be factored out, but that is rather unprincipled and would lead to the ridiculous conclusion that Turkish has no vowels because you can "factor out" rounding and backness, not to mention vowel height. You could just stipulate a specific list of things, namely those things that are notated in the IPA with a diacritic as opposed to vowel-letter choice, and knock out phonation, length and tone that way (also nasalization). That would still leave you with the ATR problem, since there are atomic letter differences (i/ɪ, u/ʊ), but these can also be spelled [i̘/i u̘/u].
Some people think that diphthongs are vowels, which raises the question of what you mean by "vowel". Diphthongs are single phonological units (at some level of analysis) composed of two vocalic articulations – two vowel segments within a nucleus. If you exclude diphthongs, that will knock out a lot of languages. I did not include diphthongs in the Dinka count (Dinka has diphthongs).
English has, in one theory of transcription, the vowels [i ɪ e ɛ æ ɑ ɔ o ʊ u ʌ] which is sort of a lot of vowels (less than German), but it can also be reduced to [i e æ ɑ o u ʌ] plus an orthogonal length or diphthongization property.
There is a behavioral rationale that people might use to justify ignoring tone, or length, when counting vowels, which is that tone and length have a certain propensity to act "independently", for example tone can spread from vowel to vowel, or may be preserved under vowel deletion (and gets reassigned to a neighboring vowel). However, the same is true of vowel height, backness and roundness. To stipulate that tone and length should be excluded (and perhaps phonation type as well) based on this autonomous behavior, while not doing the same for backness and roundness is unprincipled. There are certain strong-universalist trends in phonological theory which do stipulate that tone and length are not features dominated by the root node (whereas phonation and nasalization are), so if you make those assumptions, then Dinka has fewer vowels.
In other words, it depends on your criteria for saying this thing versus that thing are "different vowels". The probable winner is Dinka.