The answer to how many pennies can fit in one square foot (1 SF) depends on what reason is behind that question:
Let's start with a piece of cardboard that measures 12 by 12 inches, which is exactly 1 SF. As you can see in the pictures, the tape measure shows we have 12 in. on both sides (even though it may be hard to see the small numbers).
We start placing pennies on the edge of this 12 in. cardboard and it results in a perfect fit: 16 pennies next to each other measure exactly 12 inches (one foot) because the diameter of a penny is .75 in. (or 3/4 of an inch).
By the way, as you can see, we used all brand new pennies for this demonstration.
We added 15 more pennies to 'fill' the other side of the square cardboard as you can see here:
Simple math can tell us how many pennies fit in one square foot in this straight pattern (16x16=256) without actually filling the square cardboard with pennies.
We decided to do it anyway in order to show other details and for you to see how beautiful it looks. Here we go, adding row after row of 16 pennies each.
Worth mentioning here is that the pennies do touch each other. If a gap/space is required all around each penny (say for grout), changes must be made.
No penny overlaps any edge and there's no space left whatsoever - at least that's what math tells us. If you see slight imperfections in the picture below is because we placed all 256 pennies by hand and they are just sitting there (not glued).
If a penny was square shaped (instead of round) with a .75 in. side, the cardboard below would be completely covered by 256 square pennies. But because the penny is round, you can see the cardboard in between pennies.
Notice the empty area between any 4 pennies is quite big and has 4 rounded sides (more on this later).
After carefully placing every penny by hand, here it is: 16 rows of 16 pennies each.
Here's a close-up on a corner of the square cardboard. Even though the pennies are in contact with each other, there's still quite some room in between them and that's due to the straight rows pattern/layout.