Let's assume price is no object and that I'm a competent polyglot - I'm a wealthy merchant or natural philosopher or noble or clergyman, and I want to see the biggest pile of books the accessible world has to offer. Maybe there are rumors of even larger libraries in the Arab lands or the Orient, but I wouldn't feel safe or want to spend the time traveling to them and probably couldn't read the books there anyway.
I have a couple of years to spend finding the best library in Europe; where do I go?
I am afraid there are no publically accessible books at this time. Libraries are the private property of monastaries, churches, states, royals, scholars, noblemen or universities.
You can probably gain access to many of them, by getting permission from the owner or institution that keeps them. The Kalendaries in Bristol was supposedly open for the public in the late 15th century - "all that entered for instruction" might mean those that came for religious instruction though.
One of your best bets might be to go to Rome and try to get access to the papal libraries, or to the University of Paris. Towards the end of the 16th century, Holy Roman Emperor Rudolf II amassed a large library in Prague.