We thought we'd get a spare DVD player and for £17 we figured we couldn't go wrong with the Tesco's Value DVD Player. It's actually pretty good, but only does Region 2 out of the box. That's quickly fixed:

Turn it on.
Press eject (on remote).
Press Setup (so menus appear).
Press the Next button (as in, the next track button, not 'right') four times.
The current region appears at the top right (should be '2').
Press the left arrow until it says 255 (this means 'all regions')
Press Setup again to store.

Hurrah, it'll now play Region 1 DVDs too.