Access to papers as an unaffiliated individual I have graduated and my employer does not provide access to journals.
I don't mind paying, but I cannot afford subscribing directly.
Are there public mathematics associations that provide members with access to the most common mathematical journals?
 A: Yes, there are public associations that provide some access to published articles, but it would be difficult to give more details without knowing your location and field of research.
Note that many research institutes have a status of associated/invited researchers, which does not grant a salary but permits access to some of the facilities of the laboratory, e.g. email, servers, library and such.
A: Often the author has a web page on which he or she posts papers, e.g., T. Tao.
Another possible source is Archiv.
A: What I did to continue having access to articles, even tho I am no longer affiliated with my alma mater, was to apply for an alumnus ID (lifetime membership in my case!) that allows me to make at most three free(!) trips to the university library a month. I then keep a running list of articles I want to read as I encounter them (e.g. on MO or anywhere else on the Internet), and then take this list whenever I go paper-chasing in the university library. In there, they allow you to print online journals or photocopy paper journals for a very nominal fee. (It also helped in my case that I was good friends with the librarians even before I graduated.)
I've figured at this point that the price I paid for my alumnus ID, altho relatively a rather substantial sum, is way much less than the sum of the cost of all the articles I have, if I had directly paid the publishers for them.
You might want to look for a similar deal in your libraries.
