Look, I don't want to install flash or java just to find out how fast my internet connection is. I just want something that downloads a file and gives me a result. Here are some websites that do this basic thing.
For me the best one, almost magical in fact. You just click 'start test', and it works. Works in old versions of firefox without flash, and seems to produce results that agree with other tests.
2) Ten Meg
Yes, it requires you to watch a couple of images downloading and click a button when they're done, but it works in browsers without flash, java, activex, siverlight, perl, ruby, php, gtk, etc, etc, etc, etc.
3) ISP GEEKS
Described as their 'Mobile Speed Test', doesn't seem very accurate (perhaps because they're in the USA and I'm in the UK), but it gives a rough idea and doesn't need flash etc.
4) Broadband speed checker
Seems to work in I.E. without flash, though it says it would like to have flash 8. Unfortunately seems to be broken in earlier versions of firefox.
Again, not massively accurate, but simple and kinda fun waiting for the picture to download!
6) Mobile test
Aimed at mobile phone browsers, but if you click the 'wifi' option, it seems to give a fairly accurate result (but that could be a co-incidence).