Catan is a family strategy boardgame for 3-4 players (previously called "Settlers of Catan"). It has a clever modular board that is arranged differently each time you play. Catan has inspired many different 3D versions; the board represents an island with 6 different terrain types, so it is the perfect game to model, be it out of resin or cupcakes. Searching the wider web will reveal literally dozens of versions, many exquisitely modeled, and even a commercial version which sold (originally) for over $300. OK, Catan is hardly chess, but it's off to a decent start.
Note: I've written a strategy guide for the game if you're interested in learning how to play better.
The board described here was made from scrap plywood, wood glue, inexpensive acrylic paint and some leftover polyurethane.
The board that comes with Catan has hexagons and a border made of light cardboard. A few boisterous kids and a little warping and the game can be pretty frustrating to play. My initial fix was to glue the border down to a large hexagon of plywood (see pictures). Note: I recently (2015) got given a copy of the 5th edition, and it doesn't have this problem at all. There are small gaps between the tiles that prevent the warping issue.
We usually buy a family board game for Christmas, but this year I thought I'd try my hand at making a 3D Catan set out of materials I had lying around. If you want to make one like it, you'll still need a copy of the game! It's easy to find online or at bookstores.