It depends.
First, contrary to popular beliefs and the narcissistic egomaniac image propounded in the society, saying sorry for your mistakes is NOT a sign of weakness. We are all human beings and human beings err. We all mistakes. It is a sign of our humanity and it is something that should not only be acceptable, but also in fact, celebrated and encouraged. Without mistakes most of us can't learn some things perfectly.
Like T.S ELiot said, To arrive at a place of Knowledge you must go by a way which is the way of ignorance. Everyone we call experts in any field today are people who have been brave enough to try things no one has tried before, and who were not afraid to make a mistake along the way.
And this is why once a mistake is made; I mean the kind of mistake that affect a person or a lot of people around you in a negative way, the thing to do is apologize immediately. Not apologizing, in fact, is THE sign of not only weakness and cowardice, but complete stupidity.
Even the most renowned men and women in our communities have been known to apologize anytime they fall short in a way or or another. Even athletes, known for their unfettered egomanias, have been known to apologize for their mistakes, (check out Kyrie Irving's recent apology for saying that the earth is flat lol https://www.google.com.ng/url?sa=t&source=web&rct=j&url=http://kwese.espn.com/nba/story/_/id/24863899/kyrie-irving-boston-celtics-apologizes-saying-earth-flat%3Fplatform%3Damp&ved=2ahUKEwi1qcDkuerdAhWDN8AKHTlOC80QiJQBMAB6BAgJEAQ&usg=AOvVaw3s4TACbtlWf0Q3v4zsLkXf&cf=1). Hell, even presidents of countries have apologized in one way or another for one thing or another.
Now having said that,
Above I wrote that the answer to that question is that "it depends". I said that for the following reasons:
One: Incessant Apology is a madness. I say incessant apologies to mean continually apologizing for the same shortcoming over and over in an exasperating way. Nobody like this, and frankly this just shows a lack of self confidence and lack of self worth in the person doing the apologizing. If you really put your heart into an apology and show the true remorse in your heart, you don't even have to say a word, or a lot of words, to make people undertsand and forgive you.
Second, constant apologies. By this I mean making the same or diffrent mistakes over and over again, and then apologizing every time hoping this solves everything and the sky is blue and all is well with the world.
This is stupid and weak. Again this may even be on purpose and a lot of people have been known to do this to exploit some people whose meek nature makes them accept apologies easily. So people offend them continually on purpose knowing that just saying sorry all over again would make things right. No it shouldn't. This is weak and appalling.