Particle accelerators IN SPACE are the next big thing. As this paper apparently discusses, beyond 100 TeV, space-based particle accelerators become more practical than ground-based ones. As the timeline discusses, the 100-TeV Future Circular Collider will last until 2075, so the 2070s seem like a good bet on when to start building something in space. When you take into account much cheaper launch tech, it'll probably be a project on the scale of the ISS, with construction running into the 2090s before it's turned on.
As discussed here, it would likely take the form of a series of magnets in geostationary orbit. No need for fancy tubes since space is already a vacuum and it's easy to maintain cryonic temps out there.
A few hundred years further on, you could get a similar setup out in the asteroid belt, with a lot of magnets planted on little asteroids. A few hundred years after that? Well. Interstellar comets are a lot more common than you'd think. Stick magnets on a few million of those in a giant circle a couple light years across, give them some decent AI so they can figure out where the next comet even is, and you're golden.
Space is good for particle accelerators, because you can dispense with the tubes and just keep the stuff that matters.
There are also portable laser particle accelerators that are also very powerful, those are eventually going to be common in university labs.