Is it possible to create a mountain?

Theoretically, yes. Here are some people who are trying:

http://inhabitat.com/the-berg-is-building-the-worlds-largest-man-made-mountain-a-colossal-mistake/

http://www.trendhunter.com/trends/mountainous-architecture-azerbaijan-and-big-team-up-for-leisure-peaks

It's physically possible, though very costly both financially and chronologically. To start, you would need to own the land in which you plan to build said mountain on, and that alone will be very very expensive as you would need a lot of land. It's just not very ideal.

Some people can make them out of "mole hills." :-)