Well. There's a lot to consider!
In general, it depends very much on what telescope are you going to use. Go to it's webpage, read the operation manual, and check if they have the exposure calculator.
You should prepare a list of objects you want to observe, and check if they are going to be visible frem that particular telescope at that particular time. Then, use something like this
this to plot the visiblility for each object. Keep in mind that the higher the target above the horizon, the better.
Then calculate the exposure time based on object's magnitude and type. For example, if you want to get a pulsar light curve, then you'll need a lot of pictures with same exposure, so it can take a while to get one light curve out.
If you want to get a color image, then you need to use different filters for the observation, so that you'll get separate images for your R-,G-,B- channels (which also means that is takes at least 3x time to make a colour image than a to make a greyscale one).
Last, but not (oh no no no) least, check the field of view of the telescope! It could happen that the object you're trying to see will not fit in telescope's field of view! So if you want to shoot star clusters, galaxies, or nebulae, pay atention to their angular size.
As you can see, it's not that easy, and there's no answer to "How many objects it is possible to observe during one night?"
Disclaimer: I've only used a professional telescope once. It was fun, but I didn't write a proposal for it;)
P.S.: and YES, please DO post more info on what you're going to use/observe, etc.