Maybe a kiloampere?
Just for a few hundred nanoseconds though, nothing even a thin wire can't handle.
The actual DC supply is a few mA, the trick is the glass tube's capacitance has considerable peak current capacity if you provide it a low-impedance path. A typical jumper wire might be a microhenry, giving a resonant frequency of a few MHz, and a resonant impedance Zo of 10s ohms. That is, starting from a worst-case say 30kV peak, the inductance is charged to a peak current around Ipk = Vpk / Zo or maybe 1kA.
The two most important equations you need for working with LC networks:
Fo = 1 / (2 pi sqrt(L C))
Zo = sqrt(L/C)
A real set will discharge some after turn-off, as the gun doesn't cool down instantly but keeps conducting for some seconds, and the focus circuit likely draws some current from the 2nd anode (internal to the flyback transformer, which is more than just a transformer, but also rectifiers and voltage dividers, often a coupling capacitor too for dynamic focus (Trinitron) sets). A few kV is still the makings of a nasty bite, of course!
Projection CRTs I believe went to 50kV, maybe even a bit more, and had glycol-cooled faces; not to mention more xray risk (but hey, projection, easily solved). But obviously, those aren't "normal domestic".
Tim