In chapter 17, Huck meets a boy called Buck. Huck and Buck quickly become good friends, despite coming from very different backgrounds. Describing Buck, when he first meets him, Huck says that he "look[s] about as old as me—thirteen or fourteen or along there." From this quotation, we can infer that Huck must be about "thirteen or fourteen" years old, given firstly that that's how old he thinks Buck looks and secondly that he thinks that Buck is "about as old" as him.
When we are first introduced to Huck Finn, we are given to understand that he lives the life of a vagabond, often sleeping in the streets and wandering where he pleases when he pleases. We are also told that Huck wears clothes that are essentially rags. The fact that Huck Finn is only thirteen or fourteen years old makes this vagrant life of his even more pitiable. This is how we might expect an adult to live, if that adult has fallen on hard times, but it is certainly not how we would expect a child to live.
Given his life of vagrancy and the fact that he doesn't have a mother and his father is an abusive drunk, it is possible that Huck does not know exactly how old he is. When he says, in chapter 17, that Buck looks about the same age as him, "thirteen or fourteen or along there," he is perhaps guessing his own age as much as he is guessing Buck's.