Gygax shrine is located randomly in STWALT.
At Gygax shrine, you can do the following actions, namely:
a) Give Gygax an offering(click the offering action and then choose an item from your inventory), Success in this action will give you 1 xp, But most of the time Gygax stays silent.
b)Worship a..k.a pray to Gygax (20% to get 1 xp, 40% if you are a cleric. also you get an additional +10% per religious item worn such as Holy mantle and the Gygax Symbol). Success in this action will earn you 1 exp. But most of the time he will stay silent.
c) If your prayers are not answered or your offering is rejected, desecrate it. Rob and plunder the shrine, Success will result in acquiring little gold and a random item. Be warned, failure will result in getting criminal status and chances of being cursed by Gygax for 100 days!