Is it possible to objectively measure the amount of pleasure a person is experiencing? I'd like to know if it's possible to quantify somebody's "fun meter". Maybe by measuring the amount of endorphin produced. Because every person is unique, I don't expect that amount to make another person feel exactly as much satisfaction, but I'm ready to be proven wrong.

Are there any existing methods to objectively measure pleasure?
This is a very general answer to your question. And let me say upfront that I don't know about any measurement instruments for pleasure. However, I would like to emphasize that pleasure, just as about any psychological phenomenon, has to be considered a latent construct. What this means is that it cannot be measured directly, but only indirectly and imperfectly through a bunch of indicators. It is actually important to have more than one indicator in order to estimate how reliable the measurement is. Since you asked wether it is possible to measure pleasure or not, my answer is yes.  

The question then becomes what the indicators are, that allow one to measure the amount of pleasure someone is experienceing. They should be derived from a theory about what pleasure is. In the reference that @GregMcNulty gives in the comments, for instance, the amount of pleasure is measured predominantly by activity in the prefrontal cortex. But it is also stated that "From a biological perspective, pleasure is not a unitary experience in the brain. It’s not just one region that is activated when we find pleasure in something or someone." That's speaking to the notion of having more than one indicator.

Whatever way you do it, though, you will have to come up with a good theory on what pleasure is and then find indicators. And with a construct as complex as pleasure, it is important to keep in mind that you will always have measurement error.