Which planet did Galileo draw twice without realising?

Neptune. In December 1612 and January 1613 he recorded it while tracking Jupiter's moons, marking it as a fixed star - and on the second occasion he noted that it appeared to have moved relative to a nearby star. He did not follow it up.

The near-miss is more interesting than a simple oversight. He had the instrument, the observing discipline, the star charts and, on one night, the observation that should have prompted a second look. What he did not have was any reason to expect another planet: the Solar System was complete as far as anyone knew, and a faint dot that shifts slightly is far more likely to be an error than a world.

Neptune was eventually found in 1846 by arithmetic rather than by looking - Le Verrier calculated where an unseen planet must be from the way Uranus was misbehaving, and Galle found it the same night within a degree of the prediction. The gap between Galileo's drawing and that discovery is 233 years, and the difference was not better telescopes but a theory good enough to say where to point one.