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?