Some of my favorite photos are starting to feel old to me after looking at them so many times. Don't forget this is only a personal perception and often not shared by your audience.
For Nirvana, it was "Smells Like Teen Spirit." The White Stripes had "Fell in Love With a Girl." Radiohead had "Creep." All of these bands became so expected to play their breakout hit songs that they rebelled and began refusing to play them at all. There is a fear of becoming a cliche that sometimes drives people away from taking advantage of success. These world-famous bands have no problem doing something like this, but as an aspiring photographer you should be proud of your best shots and promote them regularly.
The lifespan of a photo shared on the Internet these days can be very short. I envy photographers who are able to print and enjoy their work beyond a backlight screen. What gained hundreds or thousands of likes today will be replaced and forgotten about tomorrow. Unless you are able to outsource promotion, it is your job to get your work out there in front of people. When you do this, make sure you are putting your best foot forward.
Sometimes people will only look through your first few photos so having those be your three best shots of all time makes sense. If you have trouble picking, ask a friend or loved one. If you do this you must be open to the subjectiveness of taste and preference. Don't be stubborn and force a shot that doesn't seem to move people. We all know how easy it is to ask for feedback then resort to our initial thoughts anyway.
While you should not be afraid to be loud and proud with your best shots, don't think that every photo shoot should be going for the title. Location, timing, situation, and luck all play a major role in each shot and if you only swing for the fences you'll set yourself up for a lot of necessary disappointment. Instead of always going for the next cover photo on your Facebook, try shooting with a purpose in mind. This could be exploring light, working on compositions, trying new techniques, or even just shooting for fun. You'll find your confidence and ability will grow faster with fewer expectations and more exploration.
When people complain about their computer or phone running slow, I direct them to jump into settings and lower their expectation slider. This results in mostly eye rolls, but it is true. If we have preconceived notions about how something should turn out, there is going to be some sort of contrast with what actually happens every time. If you can shift your focus to enjoying the art of capturing photos and the appreciation of your creations, I promise you can get away from the fear of oversharing a favorite photo.
I preach to share your favorites and share them often. With all the crazy algorithms out there working against everyone seeing your work, it never hurts to go back to tried and true winner.