Get Played >> How Many Plays is "Enough"
As with nearly everything else in the hobby, the question can be answered with the ubiquitous "it all depends." For me, the depending point in this regard probably comes down to overall play time than anything else.
For a light, quick-playing filler, say in the 15-30 minute range, it's likely to get more opportunities to hit the table and work its way onto the team than a monstrously deep, complex 3-4+ hour megagame. So that certainly is a factor.
We can churn out several quick games in the same session, or get it played once on several consecutive meetings, quickly inflating its numbers. Some shorter games that have become standards on my filler list include Diamant, Coloretto and Bucket King. Going just from the games I've logged on BGG (which is probably 80% accurate), I've played the above 6, 13 and 10 times respectively.
Some longer games make the cut with barely any plays -- I just happened to enjoy my few plays so much that I know they fulfill a certain gaming need and fit a certain niche incredibly well. Twilight Imperium 3rd Edition has only seen 4 full game sessions to completion, but I'm always hankering for more. I've only actually played 1 complete game of Die Macher, but I daydream about getting another game in frequently. I've also only played 1 complete game of A House Divided, almost 3 months ago now, and would love to get another game in.
With these larger games, though, scheduling is a far greater issue -- but I still consider them viable, valuable first-string options despite their lack of repeat playings (in the right context, of course). TI3 and A House Divided I don't really consider "new" any more. I'm familiar enough with the rules that I could teach the game easily after a quick brush-up. Die Macher I still consider "new" in that regard even though it's proven its worth otherwise. I've owned it for over a year, but have only played it the once, and due to the sheer density of what's going on, would need to study the rules religiously beforehand.
So when does a game stop becoming "new" for you? How many plays before it becomes second nature and you can slip into a session like a pair of comfy slippers?