Most LEGO stores sell loose bricks by the cup. I have filled many, many of these cups in numerous LEGO stores across the United States and Canada, and I have tried several strategies to maximize the number of bricks that fit into those cups. The strategy described here focuses on packing 2x4 bricks into the cup efficiently, without bending the cup too much.
Implementing this strategy involves building 5 layers that are placed into the cup with the studs to the side, instead of in their usual upright orientation. After you add each layer to the cup it is possible for fit a few small plates or tiles in around the edges if you want to.
This strategy also works well for other fairly large bricks like 2x6s or 1x8s if you build layers that are the same shape out of different bricks.
Step 1: Place 18 1x2 bricks in the grove at the bottom of the cup.
Step 2: Build the bottom layer, consisting of 22 2x4 bricks. You can also add a 2x2 brick at the top and at the bottom (shown in yellow) if you want to, and you can add a 1x2 brick at the edge of the cup (shown in red) once the layer is sitting in the cup.
Step 3: Build the second layer using 26 2x4 bricks. You can also add 6 1x2 bricks at the corners. Two additional red 1x2 bricks can be slipped in at the edge once the layer has been put in the cup.
Step 4: Completing this step will add another 31 2x4 bricks to the cup. Four 1x2 bricks can be added at the corners.
Step 5: Another 36 2x4 bricks are added to the cup in this step. The last two 1x2 bricks can be added at the corners.
Step 6: When this layer is added to the cup it will be above the brim by a few millimeters. However, the edge of the lid will still be able to touch the top of the cup all of the way around. In my experience, this has always been considered acceptable by the store staff.
Step 7: Complete the cup by using 4 2x4 bricks to fill the space inside the lid.
By following this pattern you will be able to pack a total of 159 2x4 bricks, 2 2x2 bricks and 33 1x2 bricks into a large pick-a-brick cup. While you may find other sources that report being able to fit 163, or even as many as 169 2x4 bricks into a pick-a-brick cup, the strategay presented here uses layers that fit into the cup without significantly bending it, allowing you to easily fill in the edges with whatever small plates and tiles might be available.