The proofs I know for the fact that the space $[0,1]^\mathbb{R}$ is not first countable, use the product topology in some step of the demonstration. (Reference)
So, I would like to know if this space is also not first countable in the box topology and also if there is any topology that makes this space first countable.
Edit: As Brian commented in his answer, I was looking for a topology that was different from the trivial and the discrete.
It’s even easier to show that it’s not first countable in the box topology than it is to show that it’s not first countable in the product topology: even $[0,1]^{\Bbb N}$ isn’t first countable in the box topology.
There are many first countable topologies that can be put on the set $[0,1]^{\Bbb R}$, including the trivial and discrete topologies, but I expect that you want one that is directly related to the structure of the product. Any metrizable topology will work, and there is a natural metric on it: for $x,y\in[0,1]^{\Bbb R}$ let
$$d(x,y)=\sup\{|x_r-y_r|:r\in\Bbb R\}\,.$$
