Giving people a task to work on often gets them comfortable more quickly than the random get-to-know-you games that usually get pulled out for ice breakers. Team building types of activities can make really good ice breakers as long as the vulnerability level is not too high.
One good one is to split people into groups and give each group a bag of stuff (string, paper cups, popsicle sticks, etc.). Give the groups a time limit, and tell them to build the tallest tower. Or, have a bunch of pennies handy, and have them build a tower that is at least x high and can hold the most pennies. I’ve also seen a variation with toothpicks and mini marshmallows. You have to build the tallest tower with just those two things, but you must have a marshmallow on top.
You could give each group a certain amount of tin foil and a tub of water. Their goal is to build the boat that can hold the most pennies without taking on water. When time is up, drop pennies one at a time onto each boat, and see which stays afloat the longest.
If you want to do something that’s more about getting to know each other and your group is not too large, you can have a bowl of some kind of small food (m&m’s, skittles, pretzels, nuts, etc.) that you pass around. Each person takes as much as they want, but they must take at least a certain amount (usually 3 or 5). After everyone is satisfied with how many they took, you tell them that for each piece, they must tell one thing about themselves. I’ve also seen this done with toilet paper. Each person takes as many sheets as they think they need.