Of course it's possible. Just get (buy) Microsoft Visual Studio 2013 (there are probably other compilers too) and you can write as many apps as you want. That doesn't mean any of them are sellable. The best part of writing mobile apps is GOOD DESIGN, not programming. You can be the world's best programmer but if you don't have a good design then all your effort will be for nothing.
If you want to see your competition just google for "Best Mobile Apps" and you will find a PC Mag article that indicates there are over 800,000 apps for Android just on Google Play! Do you have an idea that will put your app near the top of that list? Or even in the top 100? If you do, Great :)
You can use one of the game engines (Unity3d, UDK etc). You would still need a lot of work to do and to go into coding (depending of what you want to accomplish), but nowadays they definitelly make the game creation a lot easier and faster, even for non-coders.