How many times did Dr. Strange loop?

Unfortunately, we just don't know the answer to any of those questions, though I think we can make an educated guess on at least one.

To start with, Strange's first battle with Dormammu in the comics did not involve the Eye of Agamotto. (I believe he was actually given the eye by the Ancient One because he defeated Dormammu). In the comics, when Strange goes to the Dark Dimension to confront Dormammu, Dormammu's retaliation ends up releasing a horde of Mindless Ones (very dangerous extra-dimensional beings). Dormammu needs Strange's help to stop them, and in exchange, Strange demands an oath to leave Earth alone.

In addition, the way the montage scene is written, its clear that we're not supposed to know exactly how many times it happened; I haven't seen any leaks of early scripts of the movie to know, but I'm skeptical that they ever gave a specific count. The best we can do is just count the number we specifically see on-screen, which is around a dozen or so.

However, it does look like both Strange and Dormammu do remember each loop. It's obvious that Dormammu remembers; the whole point is that, after enough repeats, Dormammu gets frustrated and makes a deal just to escape. If he didn't remember the previous iterations of the time loop, there's no reason for him to ever get frustrated. Also, note that Dormammu tries something to kill Strange different every time, implying that he knows his previous attempts didn't succeed.

But Strange, too, changes his behavior as his loop repeats. In his opening line to Dormammu, he starts to sound bored and frustrated, which only makes sense if he knows he's done this before. Also, IIRC, he seemed prepared for Dormammu's various attacks after they'd been used on him once, which only makes sense if he remembered them.

This was discussed on the news today. There is a line cut from the movie-where Dr. Strange says "We've been thru this a 1000 times" - which would have made for a better moment if it had been left in. Dormammu would probably get that "they say our love won't pay the rent" feeling, and would definitely want the cycle to stop.

http://www.hollywoodreporter.com/heat-vision/doctor-strange-end-screenwriter-explains-twist-944334

What i believe it was an infinite loop made by Dr. Strange. If i remember he said something like this, "We can do this all the time and the world be safe", similar meaning to this.

Dr. Strange says "We've been through this a 1000 times", he made the loop so he can make the bargain with Dormammu and stop the destruction of Earth.

Considering the fact that Dormammu is being off infinite proportions and lives outside of time, pretty much a god like being, would have not yielded after any finite amount of times, they would have been at it for an eternity before Dormammu would have given up.

Any number of times less than an eternity would have meant Dormammu would have not been an infinite being as mentioned earlier in the move.

They should have emphasised that they have been at it for an eternity, they made it look like they were at it for 20 minutes or so.

Okay, I read online that a leaked transcript of this scene has been leaked. Dr. Strange mentions that they had done this a thousand times already.

So to answer your question as to the writer's idea was .. 1,000 times 10 to 15 minutes? So it really wasn't that long.