A very possible challenge. A genuine, Captain Insane-o mode challenge. But possible. Lemme see what I can do...
It's gonna take about 23000s Δv.
Aerodynamically efficient ones, I might add. 12000 more in the (very likely...) case of an unaerodynamic craft (Titans...)
No, it's not possible.
Well actually it is, but only if you're intending on making the rocket out of about 12 or more stages.
The trouble is SFSs engines. Specifically their TWR and specific impulse. These mean that in theory terms you're capped at between 5 and 6000 m/s Dv per stage (even with infinite fuel mass) but also because as the mass increases, the number of engines increases as well, there becomes a maximum number of engines you can add before they're essentially only adding weight and not relative thrust.
This problem is increased on Jupiter, where the already low TWR is butchered even more by over 2 and a half times more gravity locally.
For example, I've ran the numbers and it takes exactly the same number of Broadsword engines as it does Frontier engines (5) to achieve a 1.07 TWR for a third of the fuel mass (60t vs 160t). This is only one stage at 3500m/s as well.
With that in mind, I've done the data for the first 4 stages, with just a probe as a payload, the absolute minimum you need.
And it reads like a Rocky Horror Show. (TWR adjusted for Jupiter gravity)
Notice I'm having to use steadily more thrust efficient engines, and less and less Dv per stage?
Wanna see the show for 3500m/s for stage 3 and 4, just to illustrate my point?
That's right folks, adding another 500m/s per stage adds an extra 170
thousand tons to the launch mass.
And we're not even half way to the budget for the most aerodynamically efficient guesstimate.
Then you have to get this monstrosity (which will probably rival a Death Star in mass by this point) to Jupiter from the launch pad. A fun thought considering there's only about a dozen players on any online outlet capable of putting the smallest option 4th stage engine pack (without fuel) into LEO.