May planes be flown above the maximum certified altitude? Let's take the 737NG as an example. The 737NG has a maximum certified altitude (MCA) of FL410. 

However, looking at the MCP, the maximum altitude that can be set is 50,000ft. And the pressurization panel can be set to 42,000ft.  All allowing to fly above FL410.

Please note that I am getting this information from FSX with the PMDG 737. Though a simulator product, it claims to be very accurate model of the plane.

Also, although I am not sure, higher altitudes = fuel savings right?

So this rose the question: Can/May I fly above the maximum certified altitude? 

Can I fly safely with an 737NG at 45,000ft for example?

Related: What determines the maximum altitude a plane can reach?
The Max Certified Altitude is the highest that the manufacturer has test-flown the aircraft to. (And the 100ft simply makes cruising at FL410 legal.)

If you want to take the airplane above the MCA, and you can actually get it up there, then you now become a Test Pilot, but without the Test Pilot pay and insurance cover. Good luck!