You mean, can we measure something called the "expectation value"? Or do you mean can we measure the observable?
For the first one, we can only measure many identically prepared particles and see the "average" value as the "expectation" value. For the second one, of course we can...for that is the definition of an observable...
The thing to keep in mind is that the expectation value is just an average value. Sometimes it will correspond to an actual eigenstate (observed value) in your system and sometimes it won't. Take an electron's spin for example. When you measure, you might get a spin value of +1/2 or -1/2. Since there is an equal probability of each being observed, your expectation value of the spin will be 0. However when you actually perform an experiment to measure the electron's spin you'll never get a value of 0 because that isn't an actual allowed value...even though its the average. The expectation value can change with time or it can be a constant. It just depends on the nature of the system that you're observing. In the case of electron spin it is both a constant and not an observed value.