There's triangle below. The line is zig zags, and $P_1,P_2,P_3,...P_n$ is where the zig zags is touching the hypotenuse.
I ask, is the distance from the origin, $O$, to any $P_1, P_2...P_n$ irrational? Can the distance from $O$ to $P_n$ be integers, ect.? So I can calculate it for the first triangle, $\Delta OP_1$. The base is 0.5, and height can be found too. The answer is irrational. But what about from $O$ to $P_2$? $O$ to $P_3$? ect.
Let $x_i$ be the distance from $P_i$ to the right edge, with $x_0 = h$. Each triangle where $P_i$ is a right angle vertex is an isosceles right triangle, its both sides fraction $\sqrt{\frac{1}{2}}$ of the horizontal hypotenuse. Therefore $$x_i = \frac{x_{i-1}}{\sqrt{2}} = \frac{h}{2^{i/2}}, \quad 1 \le i \in \mathbb{N}$$ and the distance $d_i$ between $P_{i-1}$ and $P_i$ is $$d_i = \sqrt{2 (x_{i-1} - x_i)^2} = h \frac{2 - \sqrt{2}}{2^{i/2}}, \quad 1 \le i \in \mathbb{N}$$ Expanding it for odd and even $i$, via $0 \le k \in \mathbb{Z}$, $$d_i = \begin{cases} \displaystyle \frac{h}{2^k} \frac{2 - \sqrt{2}}{\sqrt{2}}, & i = 2 k + 1 \\ \displaystyle \frac{h}{2^k} \frac{2 - \sqrt{2}}{2}, & i = 2 k + 2 \\ \end{cases} \tag{1}\label{G1}$$ If $h = 1$, $1 \gt d_1 \gt d_n$, so there are no integer $d_i$ for $i \ge 1$, as OP noted.
Let $\delta_i$ be the distance between $P_0$ and $P_i$, $1 \le i \in \mathbb{N}$: $$\delta_i = \sqrt{2 (h - x_i)^2} = h 2^{(1-i)/2} \left( 2^{i/2} - 1 \right)$$ and again expanding separately for odd and even $i$ via $0 \le k \in \mathbb{Z}$, $$\delta_i = \begin{cases} \displaystyle h \left( \sqrt{2} - \frac{1}{2^k} \right), & i = 2 k + 1 \\ \displaystyle \frac{h}{\sqrt{2}} \frac{2^{k+1} - 1}{2^k}, & i = 2 k + 2 \\ \end{cases} \tag{2}\label{G2}$$ If $h = 1$, then $\delta_3 \lt 1$ and $\delta_4 \gt 1$, and $\delta_i \lt 2$ for all $1 \le i \in \mathbb{Z}$, so there are no integer $\delta_i$ for $1 \le i \in \mathbb{N}$.
If you want to examine whether there are rational $d_i$ or $\delta_i$ depending on $h$, you only need to examine $\eqref{G1}$ and $\eqref{G2}$ for all $0 \le k \in \mathbb{Z}$: for any rational $h$, $d_i$ and $\delta_i$ are irrational.