For those who don't know - Hammurabi was not only an ancient Babylon King - it was one of the first computer games ever made .  Your objective is similar to that found in Kingdom Game but you aren't playing against anyone. 

Personally, I'd rather play Kingdom Game since you are in an active world with other real people.  But, this game deserves it's credit since it's a remake of an old school original.   

This is a great strategy game in which you want to place your colored balls next to the computer's to 'infect' them.  Once you've tapped the ball you want to move the game will show you your options.  But be careful because as you convert the the balls to your color they can still be turned back.

The person to get all the balls to be their color wins. 

This is the best chess game I've seen thus far. It's very smart and will actually help you get better at chess by pointing out illegal moves.  You can also choose from various boards like wood, metal, marble, aqua etc.

All the standard chess rules apply and the computer's smart so if you want to play chess and be challenged play this version. 

Backgammon for your iPhone and it's great! This is a very slick version which looks great and allows you to even play offline.  The instructions say:

Backgammon is a board game for two players in which the playing pieces are moved according to the roll of dice. A player wins by removing all of his checkers from the board.

Although luck plays an important role, there is a large scope for strategy. With each roll of the dice a player must choose from numerous options for moving his checkers and anticipate possible counter-moves by the opponent.

Play versus your phone to get four colored coins in a row.  The computer is tricky though so use few moves and block your opponent. 

Coins can align on the vertical, horizontal or diagonal.

Jirbo.com has created a super simple version of memory which has a christmas/holiday theme.  It looks good and is easy to play and understand.  There's also a timer so you know how long it takes etc etc. 

Pretty good game from Jirbo. 

iKala is a fantastic version of the classic game. Move your beads around the board and bring them home to win. But be careful cause the application is pretty skilled!  A screenshot of the instructions are below.

Mahjong is an old school classic game. Find the matching blocks and click them both to remove the pieces.  Your ultimate goal is to get every piece removed but make sure you don't block yourself! 

Once you play this game a few times I'm sure you'll get addicted.  It's easy to understand and play but not so easy to actually win.   

This is literally Minesweeper on your iPhone.  You might remember this game from your PC back in the day.  You want to remove all the blocks by clicking them without finding any mines. 

The game is also built to increase the difficulty so if you think you're a badass go ahead and increase it!