A rough path is defined as an ordered pair $ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$ and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional information on the curve $X$.
I am not quite into their motivation, although there are some discussions online. In particular, I find the following remark in the first paragraph of chapter 9 of the book (link) `Multidimensional Stochastic Processes as Rough Paths: Theory and Applications' by Friz and Victoir: Consider $X$ of finite $p$-variation with $p\ge 2$. ... the knowledge of higher indefinite iterated integrals up to order $N = [p]$ must be an apriori information, i.e. assumed to be known.
Intuitively, I thought, as long as the curve $X$ is given, all the attached information (like $\mathbb X$) shall not be apriori, i.e. one can obtain (may be hard) $\mathbb X$ from the given $X$, which is contrary to the above.
There are many discussions on the rough path theory. However, is there any explanation on the above statement in a easier way, which can be understood to a person who has knowledge of Ito stochastic analysis but none of rough path theory?