Games that have no element of luck.
Go

Go is an ancient game; at least 2500 years old, and possibly as old as 4500 or more. Originally from China, it is now most popular, and professionally played, in China, Japan, and Korea, though there are strong and growing amateur communities in Europe and the Americas.

Go is a simple, pure abstract strategy game, with no element of luck or hidden information. The rules are quire short and simple; the simplest expression of the basic rules is 10 sentences long, though rules that cover a few extra conventions for ending the game sooner and making scoring easier, and provide more detailed explanation, are a little longer.

Despite the simplicity of the rules, it is extremely strategically deep.

The starting player has a slight advantage in Go, which would give a slight luck-based advantage to who goes first, but because winning is based on total territory scored, you can even that advantage out by giving the second player a few extra points; generally somewhere between 5 and 8 extra points, with the number having drifted upwards a little in recent years as people have discovered that the original amount was still a bit too low.

One of the great features about Go is that it allows you to play handicapped games against strong or weaker players. By letting one player start with a few extra stones on the board, you can even out a difference in skill, and still have an interesting game even though one player is considerably stronger than the other. This is great for small clubs or tournaments, when you don't have enough players of roughly equal level to compete against each other; instead you can play handicap games, and everyone has a chance to win while playing at their own skill level.

Go is a beautiful game, with a great balance between whole board strategy and local tactical battles. It has been studied deeply for centuries, and new innovative ways of playing are always being discovered. In recent years, mathematical analysis of the endgame using the theory of surreal numbers has yielded insights that have helped even top professional players, who have devoted their life to learning the game, and have teacher lineages going back hundreds of years.

I could wax poetic about Go for hours, so I'll stop here, and invite you to check it out. The Wikipedia article gives a good overview, while Sensei's Library is a wiki devoted to Go. There are hundreds of books on Go, a manga and anime series about it, professional players and teachers, schools devoted to teaching young Go players, servers for online play, Go clubs all over the place, and more.