Last week's App Smart column was all about apps for babies, because they really will love playing with your tablet and smartphone as much as you do (though you may need to pay a lot of attention lest they drop your expensive electronic item onto the floor or dribble on it).
There are hundreds of apps that are aimed specifically at babies, but don't limit yourself to these alone. There are many other surprising apps babies may love.
For example, while I mentioned a rattle app for Android (and there's a similar free one for iOS), my children loved the official Star Wars Lightsaber app $1 on iOS app when they were babies. It's simple, and it makes great sounds when you wave the phone around. There are similar free apps for Android and Windows Phone.
While still crawling, my babies also loved the kind of amazing visual reaction they could get from apps like Newton's Dream ($1 on iOS). While it's actually a demonstration of some clever physics, and technically an adult toy, in action it's a lot of brightly colored dots that move strikingly on the screen when you touch it. You may need to be present for a bit of supervision, but you'll probably enjoy playing too.
And for a strange idea, try App For Cats ($1 on iOS). It's a game where targets like ladybugs and a cartoon mouse move on the screen against plain backgrounds. They react when touched by a paw or hand. It's designed to be simple enough for your cat to operate, so it's likely to amuse your baby too.
Also don't be afraid to try apps that seem beyond your baby's age. One big hit in my home was Alien Buddies ($2 on iOS). This is an educational app themed around cartoon aliens, and it teaches abou t letters, numbers, colors and so on. While it has very sophisticated sections, my son has loved it for its fun value and pictures since he was 18 months old. It helped introduce him to puzzles.