The best - a simple pair of words - is, in my opinion, the most overhyped sentiment in American English. My favorite example of this is William Hung, the American Idol contestant whose rendition of "She Bangs" was so horrendous that people across the U.S. laughed out loud at it. Hung obviously barely practiced and didn't seek any help but still said he was happy because he did his best. His best was so underwhelming that this may seem like a joke. But he was serious. So were a lot of people who admired him for that. If you listened carefully, that night, you could hear muses from across the stratosphere saying, "That's it. I'm done."
Fortunately, the muses were still alive and kicking for Louisa May Alcott when she wrote Little Women. One of the reasons that I love the book is that doing your best is realistically portrayed a long-term action, not a single event. The characters each strove, nearly every day, to improve their work and themselves. The March sisters didn't dwell on their relative poverty. They took their jobs very seriously and talked about ways to enjoy the things they had. They also tried to do the best at their hobbies with the hope that they'd be the next great actresses, writes, artists and musicians.
The three characters with the most distinct talents are Jo, with her writing, Amy, with her drawing and Laurie, with his music. Jo, Amy and Laurie all want to be the best at their respective discipline and they work hard to improve. As they get older, a split emerges: it becomes apparent that Jo's talent for writing is unique while Amy and Laurie are seen as gifted but more ordinary in their talents. But not being the best doesn't mean that either Amy or Laurie stops practicing their drawing and music, respectively: they are still trying to improve their art, even to the last page of the books.