I was just watching President Biden signing a bunch of bills. Beside him there was a “cigar box” full of pens, from which he would use one for a bill, then hand it off to an aide before taking another pen from the box for the next bill. Is that what he is doing and what happens to the pens? Do they go to various museums or are they presented to people for different things?
They go to different people. Usually the authors of a bill will get a pen that was used to sign it into law. Things like that. If you pay close attention sometimes, you’ll see the President use several pens just for one signature.
Ronald Reagan, about to sign the Social Security Amendments of 1983:
And I am now going over and sign, and as you can notice how cold it is, twelve pens there; they’re too cold – they can only sign one letter, each pen. If my name came out to thirteen letters, I would have misspelled it.
I’ve occasionally wondered about the one letter at a time nature of signing like that. Toward the end of one’s term, it may be second nature to do that, but I’d expect the first few might be a graphologist’s nightmare.
I’m a bit surprised a practice like doing one uninterrupted signature, and however many sets of initials hasn’t been done.
Were there official, ceremonial sharpies? 
This practice had an impression on me when I was a teenager. I thought it extended to other important documents as well. I was so disappointment that I did not get a pen when I signed my first mortgage in 1965.
Did you ask them? They’d probably have been happy to give you a ten-cent pen, if it meant sealing the deal.