In STWALT, Gygax is literally the Great God of the realm.
There is a temple for (among several other things) worshiping him in the Castle; you can also worship him in the Village. Also, around the map, you can find a Gygax Shrine, wherein you can pray to him, give him an offering (wherein you offer an item to Gygax, the success of which depends on the value of the item which you are offering[successful attempts yield 1xp and the chance to be blessed by Gygax.]), or desecrate the shrine (you can get a little gold and possibly a random item, but there is a high chance that you will be wanted as a criminal or even cursed by Gygax for 100 days!)
Worshiping Gygax takes one day and has a 20% chance of success, or 40% if you are playing (and praying) as a Cleric or Paladin. This chance can be further increased by wearing a holy mantle or Gygax symbol, which both increase the chance of success by 10% each. If you do happen to succeed, you will receive one free XP. If you fail (which will happen often), he will choose to remain silent, and you will receive nothing besides that possible feeling of regret which is quite prevalent in STWALT.
Burying a dead traveller or making an offering to Gygax have a chance of giving you a temporary Blessing from Gygax if successful. When blessed, you receive a +2 bonus for all your rolls for a certain period of time. You can also purchase a blessing from the temple in the Castle for 100 Gold.