If possible, LMS and other Oxford journals http://www.oxfordjournals.org/our_journals/index.html
TAMS certain issues missing
Otherwise maybe just start from top impact factor journals.
A trick: If you look at what the library portal is doing, then you can simply modify your browser bookmark to MathSciNet as follows: http://www.ams.org.proxy.library.nmsu.edu/mathscinet/
One can make the obvious modification for other domains. The access to articles is according to your university’s electronic subscription. This appending-of-proxy approach has been successful when I’ve access other subscription sites, most especially journals!
Travelling wishlist: LMS journals, via Oxford. C R Acad. Duke. Elsevier and Springer. See AMS registry for more. DMJ100 is available at Oxford, and newer issues at Cambridge actually. DMJ100 costs 250 USD per year. Journals to fetch elsewhere: JAMS, Duke, Journal für die reine und angewandte Mathematik.
Ulf Rehmann list of digitized journals. Great page!
Intention: List relevant journals together with basic information on their scope and their electronic access policy. Check all the favourite mathematicians to obtain this.
AMS new digital directory
Apparently not online: Illinois Journal of Math (at least not older issues)
http://digreg.mathguide.de/
Google scholar!!!
Various preprint archives: Arakelov, Hopf, K-theory, Linear alg groups, arxiv.
http://www.mrlonline.org/mrl/ it seems that the articles will eventually be available
cedram.org
Russian Math Surveys, seem to be online at IOP, but no access from Cambridge
Russian Mathematical Surveys and other things available for free
Moscow Mathematical Journal. Not available in Cam. Distributed by AMS http://www.ams.org/distribution/mmj/
Canadian journals on Google books.
European Math Society Publishing House
Journal of Alg Geom: Not digitized before 2002. This is the fault of University Press, not the AMS
Communications in Algebra, available from Cam at informaworld.com
http://muse.jhu.edu/journals/american_journal_of_mathematics/
http://www.ems-ph.org/journals/jncg/jncg.php
Journal of Diff Geom: some volumes available on giga
Surveys in Diff Geom: some volumes at giga
See Kapranov’s web page for names of Russian leading journals
nLab page on 1 Notes from Glossary