"Who thought the future would be giant chemical rockets (100x BFR?)thundering around the solar system?"

It's not. The math doesn't work, the rocket equation bites too hard. This is one of the reasons to be keeping an eye on the recent claims over the past few years that fusion reactors may be something that could be shrunk down to shipping container sizes in the relatively near future. Get a functioning fusion reactor into space and suddenly the solar system is, if not your backyard, something very reachable.

But it's possible we won't see any practical way to lift off from Earth other than chemical rockets for a while.

Does the rocket equation still bite to hard if fuel production is outside Earth's gravity well?

That's why Musk had art of BFR sitting on a Jovian moon, for the same delta v to get into earth orbit, you can get anywhere else in the solar system, quick (relatively). Big rockets and large mass fractions enable all that, as my comment above indicates. Rocket equation is for real, but so are economies of scale.

There's some opinion there, but personally I would still say yes. There's a lot hiding in the "relatively" in le-mark's sibling comment.

We really need to be able to get places in a couple of weeks, and chemical rocketry just gets absurd for that; even if you can do it, you've got to fling massive refueling stations around and such. A fusion-based system would have power density sufficient to not just fly to Mars in a reasonable period of time, but to fly back without having to be refueled, which means we can go places without first figuring out how to build a fairly expensive and massive infrastructure that allows us to get back.

Even if you use fusion power you need to use some mass that you can accelerate and then expel opposite to your direction of motion. So while you would need less fuel, this is only really true if you also have what would amount to a miniaturized particle accelerator beyond anything we are able to build so far. Or put other words simply producing lots of energy is not enough.

Fusion-based rockets are still too far off. We need something now. Chemical rockets are monstrous, but that matters little once they are built and reused. Fuel is cheap enough.

If you really want something that can be put into production in a matter of years, then it would be fission-based rockets. Maybe they'll be more acceptable if they are not launched from Earth.

Once we have a real space industry I imagine there will be progress in areas like ion propulsion. You'll still need chemical rockets with >1 thrust/weight ratio to get off planetary bodies, but ion propulsion could be much better for long haul trips.

We've already had a lot of recent progress with ion propulsion. Not only are many satellites hybrid or all-electric, but NASA has launched several probes with a lot of maneuvering delta-v of ion thrusters.