"I don't have a best friend. I have a few really good friends, but no one I would say is my best friend. No BFFs for me! Is that normal?"
Not everyone has a best friend in life, and that's perfectly okay. Some people have several close friends that give them love and support, but none of which they would consider a best friend. The key is not thinking in terms of "normal," because no two friendships are alike. Some people enjoy have different friends they can go out with or talk to, while others prefer to have one person they can continually count on to be with.
It's a matter of personal preference and even, perhaps, luck. After all, you might welcome a best friend into your life, but if you simply haven't found a person that would fit the bill you cannot force a friendship into the "best" status. You should never push or hurry a friendship, so if you haven't connected with someone on a best friend level, you need to wait to see if one of the relationships currently in your life has the potential to be a BFF. If there is one, then you need to nurture and develop the friendship to see if your good friend can turn into a best friend. If you don't have anyone that comes close to being a best friend, and it is something you desire, you need to meet more friends in general to see who you might connect with on that deeper level.
However, if you are happy to have close friends (with no one that would considered your best friend), that's okay too. The key to knowing what is right is how you feel about it personally. If you feel lacking because you don't have a best friend, work on meeting more people and you'll increase your chances of finding someone. If you feel good about the friendships in your life (no matter the number), you don't need to worry about not having a best friend.