Is there a continuous map $\gamma \colon [0,1] \rightarrow \mathbb{R}^2$ which satisfies the following?
"Moves and never looks back": $\gamma(0) = (0,0)$ and $\gamma(t) \neq (0,0)$ when $t \neq 0$.
"Is never simple": there is no choice of $0\leq a < b \leq 1$ making the restriction $\gamma|_{[a,b]}$ injective.
I suspect the answer is "yes" just because otherwise would be too good to be true in the wild world of curves.