My brother and I have long discussed running a Kaiju campaign (specifically Godzilla and related). Normally I would probably select a lighter or more easily scaled system like M&M for such a game as I have had success with kaiju one-shots in the past with it, but my brother (and I) love GURPS so we would probably end up just using that as we are both comfortable with it and enjoy its quirks.
The main issue for me is not the mechanics, but the campaign structure. I don't just want to run a series of giant monster battles as I feel it would essentially become a strategy board game at that point. I also don't want to run human-only where the players are scientists, soldiers, whatever and basically just have to deal with the equivalent of natural disasters every couple sessions.
My idea is this: each player would have a human (or alien, or whatever) character to play by default. They would likely be a part of the same organization, or closely related to each other in any case. The bonding element would be that these characters would all be psychically sensitive to monsters (ala Mothra's twins); I hadn't decided if they would pick which monster they were 'bonded' to, or if it would be a general sensitivity.
In any case, as problems crop up and the monsters do their thing, characters would be able to make the equivalent of an empathy check or reaction roll towards their/a monster. If successful, they get to temporarily "control" the monster as their human gains empathic communication with the creature. Fail, and the monster either does its own thing/ignores the person, or possibly reacts very poorly and does the rampage/destruction thing. There could be enemy espers as well, vying for control of the monsters for their own nefarious purposes.
This way, you can play both the human elements of the campaign as well as play the big monster fights, and ideally would link the humans to the monsters in an interesting enough way that it makes the impersonal monster/human elements slightly more dynamic, plus create some interesting tensions with regards to conflicting goals and motivations between the humans and monsters.
Anyway, I've never set down anything firm on it, but this to me seems like a plausible way to run a sustainable campaign around such a premise. An alternative would be just humans in mechs fighting giant monsters, but I'm less interested in such a premise and I feel like playing different sorts of kaiju at least some of the time is half the appeal. If I ran a human-mech game I would vastly prefer it be mech on mech violence, either in a military context or something like Robot Jox where it's half game/sport, half war.