While the first day and week of school is great for getting to know you games and activities, this is work that can and should continue throughout the year. Building a community in the classroom takes time, and doesn't end after the first crazy week of school.
Below are some getting-to-know-you and team-building activities that you might try in your classroom, even though we're beyond the first week of school. Though you are waist-deep into curriculum now, you can make these fit your content needs so you can not only meet standards, but have fun getting to know your students too.
Time Capsule Activity: Have students divide a page into three columns. The first column can be "5 things I know already," the second can be, "5 things I wonder about," and the third can be, "5 things I hope to accomplish." Now that they've had some time to learn about grade X or your content, they're more likely to have informed "wonderings" and accomplishments to think about. Have students share with partners or small groups, and then keep these to hand back at the end of the year.
Snowball Fight: This is a more active version of the previous suggestion. Students draw 3 columns on a plain piece of white paper. In the first column they write what they are excited about, the second is for what they are nervous about, and the third is for something they want to learn. Students crumple their paper into a "snowball." Divide the class in half and students toss their snowballs to the opposite side of the room. Students take a snowball and try to find their partner.
I Am From: One of my favorite getting-to-know-you activities is having students write "I Am From" poems based on George Ella Lyon's famous poem. There are templates out there to use, but I tend to like giving students freedom to really play with the style and make it their own. You can read about an example of teaching this poem structure here.
If You Build It: A classic problem-solving / team-building activity where students are given a set of materials and need to build a boat/castle/table in a given time period. Student teams can write about their process and then also write about what they would change for next time.
Worst-Case Scenario: Students are given a scenario (stranded on a deserted island, lost in wilderness) and come up with a plan in which everyone safely gets home. They can be limited to 5 items they have with them (and you can nix certain things like planes or time-machines :) Students can vote on the best solution.
I've written about building relationships here and here for "getting-to-know-you" activities. This is a year-long process, not one that is done the first week or month of school. I always love that I can learn new things about students the last week of school, that they can surprise me even then. Allow yourself that surprise and keep getting to know your students all throughout the year.