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The Nintendo DS with Opera is now the killer handheld web browser, it's incredible usable
I find it painfully slow, and the GUI is very unresponsive when the RAM usage is high (which is most of the time as the DS only has 4MB internal + IIRC around 8MB extra with the expansion slot). The GP2X would be a lot smoother web browsing experience, although I can imagine the battery issue. How long would your fully charged Tungsten T3 last without it?
What it could do is authenticate, download the html page + pictures the html asks for (or maybe just cut pictures altogether), then disconnect until the next page request is sent - that would save quite a bit of battery if for example you were reading an article or a long email on the browser.
Then again, web browsing isn't particularly the most cpu straining process, underclocking could be done to reduce more power.