Fun to shoot means that it is both accurate AND that you can hit with it. For instance, a 4 pound 50 BMG might very well be accurate, but how well you gonna shoot it?
Second is that it has to have some power and recoil, but not too much. BB guns are fun to start with, but no one sticks with them over rimfires. At the same time, you don't want beat-the-snot-outta-you recoil. Just a gentle rap or push is enough.
Also pertinent to this discussion is the cost of ammo. For instance, a 45 LC may be fun to shoot, but unless you reload for it or are undeniably wealthy, you won't be able to afford to shoot it more than twice a year.
Finally, my own fun guns include my SIG P-225, CZ-452, and Bulgarian AK. The Finn M39 is also high on my list of fun guns, at least shooting kneeling or offhand. From the bench, it ain't quite so fun.