How big can a gas giant become before it's considered a star?
How does the theoretical maximum compare the observed maximum?
You mean like Arthur C. Clarke's 2010 when Jupiter turns into a star? We often turn to Jupiter's mass ($M_j$) when thinking about this problem.
It turns out there's a whole class of stars that fuse so faintly that we can only see them well in infrared. Brown dwarfs (which are still called "stars") turned out to be so cool that only new infrared technologies could find them. We now know they are very common, so common that new classes, L and T (cooler than M) had to be made for them. Surprisingly they turn out to be about the same diameter as Jupiter. Between 0.073 solar masses (78 Jupiter-masses) and 13 Jupiter-masses, brown dwarfs do fuse their natural deuterium (heavy hydrogen, with an extra neutron) to helium. Below 13 Jupiters (0.0124 solar masses), fusion stops altogether.
The brighter stars like our sun begin above 0.073 solar masses where they are hotter and emit more visible radiation.
So you need at least 13 Jupiters to get it going and the theoretical limits are still being refined by observations of Brown Dwarfs. There is a fussy line between planets and brown dwarfs. Small brown dwarfs can still be considered stars and not planets even if they are not fusing because they probably burned off all their deuterium (form of hydrogen).
Currently, the International Astronomical Union considers an object with a mass above the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 Jupiter masses for objects of solar metallicity) to be a brown dwarf, whereas an object under that mass (and orbiting a star or stellar remnant) is considered a planet.
The 13 Jupiter-mass cutoff is a rule of thumb rather than something of precise physical significance. Larger objects will burn most of their deuterium and smaller ones will burn only a little, and the 13 Jupiter mass value is somewhere in between. The amount of deuterium burnt also depends to some extent on the composition of the object, specifically on the amount of helium and deuterium present and on the fraction of heavier elements, which determines the atmospheric opacity and thus the radiative cooling rate.
Like what was said above, the smallest star is a brown dwarf, at 13 times Jupiter's mass; at that mass, deuterium can fuse. As for observed results, the most massive planet ever discovered is DENIS-P J082303.1-491201 b, with a mass around 29 times that of Jupiter. Because of how planets form as compared to stars, planets probably can't get much more massive than that.