I had a quasi-brush with the Galileo, although not in "person".
Around 1986, I was president of a sci-fi club at the University of Florida, and we started planning a convention to take place on campus in (IIRC) 1987. It was to be named "GatorCon". (Not to be confused with one by that name, organized by others, that actually took place many years later.)
I don't remember how he found out about us, but a fellow called me up to offer the Galileo for display at the con. He was a photographer who lived near Jacksonville, FL, and had nurtured, as it turned out, a bit of an obsession concerning the shuttlecraft for many years. He had visited it and videotaped it back when it was still rusting away on the Paramount backlot (EDIT: actually in a private open storage lot, probably during its time at Rebel Storage) many years before, and had more recently gone to see it and its current owner after its 1980s restoration. He didn't actually own it but was very near to acquiring it and buying a Ford Aerostar van (because it "looks like the shuttle") to haul it across the country.
I certainly thought it would've been awesome to have at the con, so I agreed to have him come talk to the club about it at one of our meetings. He seemed a little odd on the phone, but enthusiastic.
So he comes, and gives me 8x10s of the bird in its current condition (see below), and shows us the video he'd shot when it was still decrepit. He had edited it into the scene when Kirk approaches the Enterprise in the pod from TMP, replacing all the shots of the Enterprise with his handheld videotaped shots of the shuttle. They even kinda sorta matched, in that he circled the ship at the appropriate times, and trucked into the hatch when the pod was docking. It was all we could do to keep from laughing at the very idea of the video--here was Kirk just MARVELING at this rusted out hunk of crap--and the fact that he'd obviously put a lot of work into something so stupid. Not to mention the amateurish execution (early camcorder tech). He was even nerdier than US! But hey, at least it showed passion.
But we couldn't accept his proposal. As cool as it would've been to have the shuttle present, his terms were out of the question. I forget the specific number, but he wanted something like $5,000 for the appearance. Whatever the figure was (below five figures but several grand for sure), it was about what he was going to be paying the current owners for the thing (with the cost of a trailer for it added in, IIRC)! In effect, he wanted us to buy the shuttle for him.
Fat chance. That figure was several times more than what we'd been budgeting to pay guests with actual heartbeats. There was no way, financially nor ideologically.
He never did acquire the shuttle. As for our con, it didn't happen. Although our faculty advisor had experience with event planning, we were way over our heads and canceled it. This, however, was after I'd already notified STARLOG magazine about it, so it appeared in their con schedule column. Reportedly a couple people actually showed up looking for it.
The experience did facilitate me corresponding very briefly with Majel Barett Roddenberry, though, so that was cool.