I suppose about 8 or 9 before the whole thing about several copies of the same person got weird and freaky.
How many Sarumen would be necessary to make you run back to Hobbiton?
3 at the most, in part because I’ve seen too many Chrsitopher Lee movies.
How many Jay Gatsbys could you handle before deciding that F. Scott Fitzgerald is an evil man?
I’d say ten or so. I rather enjoyed Mr. Gatsby.
How many of those Kings off a deck of cards would freak you out?
Just the one with a sword through his head. Nearly makes me puke every time I see that massive head wound.
How many paranoid androids would it take to make you crave the solace of the Total Perspective Vortex?
I think three would do it. Two of them would just hide from each other assuming the other mean it harm, but three might form a team.
How many Members of the Syndicate would it take to scare you into looking for FBI help?
paranoid androids
If it is Marvin’s room, then not even one of them is needed; those tidal waves of depression he keeps in there will kill you in less than a minute.
Outside that room: 3 would be enough to send anyone to despair.
I see on preview that elfkin447 beat me to the exact number I was thinking!
Is that a “The Wire” reference? I don get HBO, I would guess just 1 member.
How many Tarzans in a room?
No, not a reference to The Wire. I bet someone who knows what I meant will be along. (though the link in my sig…)
Depends on which sort of Tarzan you mean. Disney animated Tarzan 4 or so, because I’d worry about them forming a pack. The WB’s Tarzan, just the one. He’s a poor conversationalist and keep jumping through (closed) windows.
How many Fraggles would make you drop your doozer sticks?
If we’re talking Henson creatures, then just two of them would do it. Anything with a hand that far up their ass really creeps me out.
How many Dirk Diggler’s would it take to increase your pucker factor?
Probably four, at which point they’d be too close for my personal comfort.
How many Nick Carraways could you handle before wishing that he stayed in Minnesota?
OT: You know, Gov, I find your Gatsby obsession a little disconcerting. Could you go back to '50s politics, pls.? 
/OT
He’s such a nonentity, I think I could go into double digits – let’s say an even dozen.
How many Barneys could you take? (Rubble or dinosaur, your call.)
Bah. I could take at least 20. He doesn’t really do anything, after all.
How many Evil Ash’s would it take to send you running away from that damned book?
Just the one, due to my dislike of horror films.
How many real people in films played by other people (E.g., Larry Parks playing Al Joelson) would it take before you get terminally confused?
I think 157 would be enough. I usually do ok, but 157 would be like the black hole of Callcutta.
How many Chers from Clueless would be so many you like totally couldn’t take it any more and stuff?
I think I could be piled on by about thirty or so Chers before I’d insist on coming up for air.
How many Heathers would it take to make you fear for your life?
3, and I think you know why.
How many ** ALFs ** would it take to make you feel like you’re not really number 1?
I hate the one enough as it is.
How many guys in red shirts from Star Trek will it take before you want to say “Bones”?
Oh, it takes a lot of those. Especially since wearing the red shirt seems to mean inevitable death by the end of the episode.
How many Cyphers would it take before you wished you took the blue pill?
6.7 Cyphers for me, no more.
How many Lawnmower Men would it take to make you feel like floating, falling, or flying?
One story lawnmower man, four crappy movie lawnmower men.
How many Smurfs would it take before you got that not so smurfy feeling?
1 Grandpa Smurf, or 4,000 Smurfettes if I get my pick.
How many droids would it take to hornswoggle you into an intergalactic adventure?