How would I make a layer wrap around the other side like Pac-Man?

You can add simple modulus expressions to reset the positions every time they reach a certain value:

mPad= 50; //invisible padding

mX=thisLayer.transform.position[0];

mY=thisLayer.transform.position[1];

cX=thisComp.width;

cY=thisComp.height;

fX=linear(mX%cX,0,cX,-mPad,cX+mPad);

fY=linear(mY%cY,0,cY,-mPad,cY+mPad);

[fX,fY]

Mylenium

I think the Offset effect can achieve this.  You'd animate the Shift Center To property instead of the layer's position, but I think it's the simplest way to do this.

The easiest way to accomplish this is to animate your layer from one side of the screen to the other, starting with the layer's in-point. Trim the layer as soon as the animation leaves the other side of the screen. Now, Pre-compose the layer, moving all attributes to the new comp and trimming the comp to the layer duration. Next, apply Time Remapping, use the K key to move to the last time remapping keyframe, then move back one frame (Ctrl/Cmnd + back arrow), set a new Time remapping keyframe at the start of the last frame, then select the last keyframe and delete it. Now add a simple loopOut() expression to time remapping.

You can now extend the outpoint of the Pre-composed (nested) comp, and the layer will loop as long as you want it to loop. You can also change the speed of the move by simply sliding the last keyframe to the left or right without worrying about jerky motion or duplicate frames.