Great suggestions!

As far as I know at my own university the official alumni organisation provides alumni with the ability to VPN/proxy over the university.

That's a prety good investment if I can enroll at a university that offers VPN for alumni. My university doesn't let alumni access the network, and I think from a quick search that most US univerities don't because of license restrictions. I'll check out universities in other countries.

http://www.deepdyve.com/ is also worth a look if you don't have access to a university. A free account allows you to view journal articles for five minutes. There also a 40$/month professional plan that gives you longer access to 40 articles per month.

Nice! This will be useful right now, so thanks for mentioning it. Unfortuntely, their journal selection is limited compared to a university library, and paper downloads are only 20% off the publisher price (and limited to 40 papers per month). I think I'll try contacting them for custom bulk download plans.

You could also pay a student to be able to use his VPN. I don't know the legalities of it. It might be illegal. There might also be different laws in different countries.

Account sharing is not allowed at my university, and I think most schools in the US don't allow it.

http://www.reddit.com/r/scholar is a source where you can ask for specific journal articles. But I don't know the legality of the endevour.

There's also the Less Wrong help desk. Both are useful, but it takes time for a person to process requests, and neither are suitable for high-frequency requests.