I have used all the four ports on my USB hub, a dongle that plugs into one of the integral USB ports. I could buy a bigger replacement, but what if I buy a second hub of the same size and plug it into the first? Would a daisy chain of USB hubs have any problems?

If your USB hub is not powered (runs off of the USB port power) you may overload by daisy chaining. Get a powered hub, and use that as the first hub. You can hang a simple hub off of the powered hub without overloading.

Note: the performance may suffer if you hang too many devices on a single USB port. Generally, you can only count on the performance at the speed of the slowest device.

If you have a hub with external power, you should be able to hang another off of it just fine. Again, better to hang it off of a different USB port if one is available, but it should work.