Games for a Birthday Party

Set up an obstacle course. We did this for my niece's birthday. The kids threw a pin pong ball through a hanging hula hoop, ran back and forth through a line of cones, crawled through a tube (the collapsible kind that come with kids' tents), walked a tightrope (rope on the ground), and fed an elephant (tossed a peanut on a chalk drawing of an elephant). We set up two courses so the kids could race. We raced twice and luckily each team won once. After that the kids kept running and running and running the course. The little ones could do it and the big kids ran it too. They loved it.

Hot potato/pas the parcel. Wrap a gift in lots of layers of paper. When the music stops the kid holding takes off one layer. Repeat till everyone has taken a turn unwrapping when the music stops or you have an unwrapped prize...
Also musical statues or chairs - freeze/grab a chair when the music stops

We used to do the marshmallow on a spoon race - that's fun... We always do decorate your own cupcake - this eliminates the need for a birthday cake. Stick the nose on the snowman or pin the tail on the donkey are always fun...
We decorate our own goodie bags, too...
YMMV
LBC

When my daughter turned 2 we did the hokey pokey, and I made a pinata. Those were the only 2 planned activities. For foods I served all finger foods...fruit, cheese, crackers,and I made mini cupcakes rather than a cake. It all worked out great! When she turned 3 she wanted a princess dress up party.. we played pin the crown on the princess (a life size poster of her dressed as a princess) and blindfolded the kids with a sleep mask.

As a child I remember playing a game where we dropped knelt on a chair and dropped clothes pins into a milk bottle.

A simple craft may also be an option...or having someone do face painting...that was a big hit at a recent party we attended for a 3 y.o.

Freeze dance. Even the littlest ones can play. Musical "squares" where they have to find a square instead of a chair when the music stops (you don't even need to take any away - they'll just have fun running to the spot if they're 3-4 years old). Also, not a game, but our kids have always had fun cookie decorating (frosting, sprinkles, etc) but that does get messy fast. Have fun!