Should you ask for recommendation letter from the professor of a class you didn't get an A in? It seems like the norm of American universities is that all the letters of recommendation are always positive stuff, and they are normally evaluated on the level of specificity they have, and if they have something extraordinarily positive about the student, rather than some usual positive language. Given that they never write a negative letter, would it help my application if I ask for a letter from the professor of a class that I didn't get an A in, if the professor is a really big name in the field? 
(I got an A-, so not so terrible, but the grades were curved and I still got an A-.)

I was also wondering if the answer to that question would be different for applying to European universities, given that their expectations from a recommendation letter seem to be different, and they don't necessarily expect a personal description of the students' abilities.

(The professor is a really really big name. He is one of the 3 most famous people in the field in the entire world.)
I think there are two questions here that help determine whether or not you should ask this person:

Do You Have An Alternative? : This may seem like a simple question, but if this is the only person who you've done relevant coursework with, etc. you might not genuinely have a choice.

Can They Speak to Your Abilities? : While in the United States letters of recommendation often contain positive comments, they don't have to strictly be about people who are unalloyed perfection. What is important is that the person be able to speak to your abilities as a scientist, your probably success in graduate school, etc. "Merely" getting an A- in their class might not mean they don't think highly of your ability to think critically about the subject, etc. - just as getting an A is not a guarantee that they do.

For reference, I ended up looking back at the people I had write my letters of recommendation - I had taken classes with two of the three, and I had gotten A's in neither class.