Typically containers get heavier as you add to their contents. One notable exception to this is a USB memory flash drive.
USB drives store data in binary form, using ones and zeroes on transistors. When you save data, a binary zero is set by charging the float gate of the transistor, and a binary one is set by removing the charge.
For each charge, electrons are added, tacking on 0.00000000000000000000000000091 grams per electron.
An empty USB drive holds mostly zeros. This means that an empty drive weighs more than a full USB drive.
Don’t worry about pulling a muscle by having to carry a full USB drive, though. It would take more USB drives than currently exist before the difference in weight would be noticeable.