I live in a very small, old house with a hip roof. But it still has 11 planes: 4 for the main part of the hip roof, 4 more for two attic vents, and then three more for the little entry (also hip roof style).
Did either of you find out which roofs (rooves?) were naughty and nice?
Okay, I’ll play too. How many planes can you count on this roof?
(From this page, with additional pics:)
Re: curved rooves?
Or zero. OP explicitly asked for count of “flat surfaces”:
Flat roof, so either 2 or 3 depending if you count roof sections on the same plane but interrupted by more structure in the middle as 1 or 2.
My house, 8. My neighbor’s house, 13.
We live in a neighborhood of 1920s mock Tudors. The builder apparently had a thing for gables and turrets, and both of our houses have additions. They’re smallish houses, around 1600 sq ft
Might win–the main roof is one plane – a hyperparaboidal roof–goes from 1/4"/12 to 3/12 in 80 feet–the other roof is a french curve with 36’ radius up, then a 36’ radius down. I’ll try to post a pic. Not sure how, though.
Five. A “hip” roof with an attic dormer window.
Six. A gabled house with a gabled garage joined by an extremely short gabled extension between house and garage. Garage roof is at 90 degrees from house and extenstion.
My former house was 108 feet long and with a hip roof, four planes over 3000 square feet of house.
Three. The original part of the house (built 1880) has a classic box gable style roof, then there’s a flat roof on the kitchen extension at the back.
Two: my house is a boring 3 BR one story rectangular ranch. Open gable, ridge on long dimension.
Brian
4: open gable, with a covered porch across the front, and a flat roof over the kitchen addition in the back.
Just 2. It’s a gable roof not pitched nearly enough to account for the amount of snow we get.
Just 2. It was built some time between 1900 and 1924. It’s pitched far steeper than a central valley California house needs to be. It may have seen snow in 1976.
- Back, front, and bedroom over the garage.
I’m guessing one? It’s an apartment building, so I’ve never actually seem my roof. For all I know it could have a dome.
Actually, none. There is a big swimming pool up there. Unfortunately, only the cool tenants get to use it! 
Too many. I have a Cape Cod style house, with a 45-degree on front (that I don’t like because it’s so hard to work up there) and about a 20 degree on the back. The dormer windows have roof on top of them at a different pitch.
Then there is a new section built onto the right with a kitchen and living room, it has 3 different pitches, about a 20 degree on the back and front, but increases to 45 about halfway up. Then there is an attached garage to the right of that with a 45-degree on each side. And the dining room in the back has a bay window with a sort of hexagon-shaped roof over it.
So like 11 at least.
I think I have found a winner. This house in Summit County, Ohio has 20 hexagonal turrets plus other roof planes. So a minimum of 120 planes. I’ll let someone else get an exact count. Not too far from me. I may have to took a look some day.
Only two. It’s a 2 story split with a “smaller” second floor roof.
On a bit of a different note, we live in a residential area where the houses are 35ish years old. When Mrs Mallard and I go for walks, I always look for roofs where the shingles are curling and need replacement. (and no, I’m not a door to door salesman although a roof salesman could probably make a fortune in our area) 
Mine is all broken up over different levels… I think I count 7 planes, but some of them are pretty oddly shaped when they wrap around a lower level.