I have never been into the "shoot, bang, blood and guts" games that others like. I am more the, solving puzzles and building worlds, type of gal.
I started a few years ago with MYST...
....where you have to grab the key, to unlock the door, to get the secret book to save the world. You can't die and there are tons of puzzles and mysteries to solve.
Then I found BIG FISH GAMES...
....and I became addicted to hidden picture games. You have to find objects in the scene that help you find clues to solving a mystery.
This screen shot is from ESCAPE THE MUSEUM 2. You have to find all the items in the menu on the right, which will help you do other things in the game. They are really quite fun.
The most recent game I downloaded is called WORLD OF ZELLIANS-KINGDOM BUILDER.
It's an adventure game. You have to help different kings build up their kingdoms before fires, thieves and crazy people take over. There are over 30 levels and it's really quite challenging.
I also play FARMVILLE on FACEBOOK. Let me know if you want to be my friend or need me to send you chickens or barrels of apples.