Throughout the Star Trek franchise, we are shown that the Enterprise is held in especially high esteem within Starfleet. For example, Riker refused his own ship to pursue a tour on the Enterprise, and it seems that not many (if any) other ships have been recommissioned again and again so many times (Enterprise-D, Enterprise-E, etc). It is often the flagship of Starfleet, and many crew members are unusual or highly respected (such as the unique Data or Captain/Admiral/Captain Kirk). And of course, the adventures and exploits of the Enterprise are legendary, with entire wars and even threats to reality itself being thwarted regularly, and many revolutionary discoveries being credited to the ship and her crew.
However, we've also seen enough of the larger universe to know that other ships get into similar situations. The Voyager crew, for example, got into quite a few hijinks, and the crew of Deep Space 9 didn't have an easy time of it either. It's probably safe to say that Sulu's Excelsior had an interesting service history, from what we see of it, and every once in a while we encounter a Starfleet officer who isn't terribly impressed by the Enterprise, implying that it's not objectively unique, or at least not indisputably so.
Are we therefore to assume that most, if not all, Starfleet ships have similar adventures? That encounters with extra-dimensional beings, ancient planetary disputes, accidental time travel, and threats to the fabric of reality are the norm, or at least not unheard of, for all major space-faring vessels? To put it another way, is that just how things work for a galactic civilization, just as jet lag, juggling time zones, and international law disputes become commonplace for a planet-spanning society?
Or do the adventures shown in the various Star Trek shows and movies constitute the "best and weirdest" of the things that happen in the Star Trek universe, and the Enterprise (and Voyager and DS9, etc) are unique in the quantity (or sheer existence) of unusual "adventures" among the ships of Starfleet?