It might seem silly, but my favorite animals are chickens. Not only are chickens pleasant to be around, they're useful in keeping the bugs down. Slugs and worms are on the menu, but so are grass, weeds and garden plants, so you have to be careful about where they're allowed to roam.
Not into chickens? Not everyone is, but even if you don't enjoy them as much as I do, you still might be able to benefit from having a few in your backyard. They may not be as productive as this article says and they may even be more trouble (think: your neighbor's dog breaking into your yard to chase your chickens!). Hens will chatter softly for you, and if you're around them often, they can be quite friendly.
If not chickens, how about rabbits? There are people who have them in hutches in their backyard. Rabbits are quiet and fairly easy to care for, but you must know a few things before breeding them - like whether to take the doe to the buck or vice versa. They're economic sources of protein since they proliferate and grow quickly.
If you decide to try one of these, be sure to check with your city. Not every place allows for animals other than pets, although laws are gradually easing. Growing your own protein, whether it's eggs or meat, and taking part in natural animal husbrandy, will enrich your life and could make a nice dent in your grocery bill, too.
Images courtesy of morguefile.com