Thick as you can make it and still make weight!

Your front and back wedges should certainly be metal - toughened steel is your best bet, such as hardox which is pretty popular with robot builders. If you're going to enter events with spinners (the uk championships and uk tag team championships) then you will want something a fair bit beefier than HDPE - hardox, grade 5 titanium or thick 6061/7075 aluminium (and by thick we're talking 20+mm) to hold up to some of the spinning weapons going at the moment.

If you aren't worried about spinners you should be able to get away with 10mm of UHMWPE (tougher than HDPE, bit more expensive though).