How many planets were there in the Empire at the time of A New Hope?
Must be a real planet or large moon
Must be inhabited by sentient beings
Must be governed by the Palpatine's Galactic Empire.
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According to Wookieepedia
The Galactic Empire's territory at its peak consisted of some one and a half million member and conquered worlds, as well as sixty-nine million colonies, protectorates and puppet states spread throughout the entire galaxy, stretching from the borders of the Deep Core to at least Wild Space.
Note the distinction between world and colony - I assume 'one and a half million worlds' match your 'real planet or large moon' requirement.
There is a map of the Star Wars galaxy which claims "50 million inhabited systems". Assuming that most of these systems are simply 'colonies', the Wookieepedia claim of 69 million colonies could be borne out quite easily. The map also reckons that there are approximately 10^17 inhabitants (or about 2x10^10 per system by my rough reckoning - by comparison, Earth is currently populated by just short of 7x10^9 or about 1/50 of the average Star Wars system). I'm not sure of the canonicity of this map, though, and it isn't clear if this is the population of known space, or the Empire alone.
Another source is the Star Wars: Technical Commentaries. It does appear to draw from various RPG sourcebooks and EU novels. This claims 12 million worlds - possibly including colonies as worlds, but this is far short of the claims from the other two sources. It also indicates possible discrepancies in various canon sources:
The galaxy successively governed by the Galactic Republic, Palpatine's Empire and the New Republic was a typical spiral galaxy consisting of several hundred billion stars. According to Lando Calrissian in Shield of Lies (p.39) the galactic disk is 120,000 light years in diameter. According to the musings of Boba Fett in Tales of the Bounty Hunters, there are of the order of a hundred billion star systems and twenty million intelligent species. A hundred billion star systems is typical of a galaxy of this size, and the total number of stars would be somewhat larger since a majority of stars are in binary or multiple systems.
According to Dark Empire there are over twelve million inhabited systems. Most other references are less specific, including only vague notions of "a thousand thousand worlds." In agreement with Grand Moff Tarkin's statement about the Empire's membership [A New Hope, p.116], The STAR WARS Roleplaying Game Second Edition [p.126] indicates over a million "member worlds", supplemented by "colonies, protectorates and governorships" amounting to "nearly fifty million systems." The member worlds presumably are the only ones directly represented in galactic institutions. The book also states that the total sapient population is only about 100 quadrillion beings, meaning 10^17 or 10^26 depending on whether the colloquial or traditional interpretation is used. This seems remarkably low (and is perhaps unrealistic) considering the number of worlds, and the resources of barren systems that can feed them.
I realise that I haven't provided a definitive answer to your question - but these are the best sources I could find for you, and without something being mentioned in a movie, this is what we're stuck with.