Let $\gamma: [a,b]\to\mathbb{R}^d$ be a smooth (i.e. $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve with arc-length defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)).$$

Assume that $\gamma_j(t),\gamma_j^\prime(t)\neq 0$ for $1\le j\le d$.

Define a possibly transfinite partition $P=\{a,x_0,x_1,\dots,b\}$ on $[a,b]$ by: $$ x_m= \max \{t\in[x_m',b]\mid \forall s\in [a,x_m'],\gamma_j(t)\neq\gamma_j(s) \quad \forall 1\le j\le d-1\}, $$ where $x_0'=a$ and $x_m'=\sup\{x_l\mid l<m\}$ otherwise. Roughly speaking, this partition divides $\gamma(t)$ to injective graphs in respect to last coordinate. Let $\Omega$ be the order type of the set of $x_i$, that is, $x_m$ is defined for $m<\Omega$ but $x_\Omega$ is not.

My question is "how large can $\Omega$ be?". Clearly, $\Omega<\omega_1$ since there is no order preserving embedding of $\omega_1$ into $\mathbb R$. Can we find a curve $\gamma$ with $\Omega \ge \omega$? Can we bound this ordinal?