Let $W(\omega,x):\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Brownian motion where $\Omega$ is the sample space. Recall that the quadratic variation of $W$ over the interval $[a,b]$ equals $b-a$ almost surely, that is to say:
$$\langle W \rangle = \lim_{n\rightarrow\infty}\sum_{i=1}^{n} \left| W\left(\omega,a+\frac{i}{n}(b-a)\right)-W \left(\omega,a+\frac{i-1}{n}(b-a)\right) \right|^2 = b-a$$
almost surely.
Now consider some continuous function (this time not a random variable) $f:\mathbb{R}\rightarrow\mathbb{R}$ and consider the more general limit:
$$\langle f \rangle = \lim_{n\rightarrow\infty}\sum_{i=1}^{n} \left| f\left(a+\frac{i}{n}(b-a)\right)-f \left(a+\frac{i-1}{n}(b-a)\right) \right|^p $$
I would like to know if there exists a continuous function $f$ for which $\langle f \rangle$ is finite but non zero for a value of $p$ greater than 2. For example, does there exist a function $f$ of finite but non zero 'cubic variation' ($p=3$).
More generally, for which values of $p$ do such functions $f$ exist.
The paths of fractional Brownian motion with Hurst exponent $H$ admit (almost surely) non-vanishing $\frac{1}{H}$-variation. Since $H \in (0,1)$, then you can find paths with your desired property for $p = \frac{1}{H} > 1$.
But, of course, that requires a probability model on path space, which is not what you want. We can still provide explicit paths that admit non-vanishing $p$-variation for any $p>1$. For simplicity, I'll change the partition on which you sample points on your path to the dyadic partition of $[0,1]$. We define the Takagi-Landsberg functions with Hurst parameter $H\in(0,1)$ by $$\mathcal{X}^H= \left\lbrace f \in C[0,1] \, : \, f = \sum_{m=0}^\infty 2^{m\left( \frac{1}{2} - H \right)} \sum_{k=0}^{2^m - 1} \theta_{m,k} e_{m,k} ,\, \theta_{m,k}\in\{-1,1\} \right\rbrace$$
Where $e_{m,k}$ are the Faber-Schauder basis functions for $C[0,1]$. Then, by Theorem 2.1 of [1], we find that if $f \in \mathcal{X}^H$ then the sequence of functions
$$\langle f\rangle_{t}^{(n)} = \sum_{\substack{k=0 \\ k2^{-n} \leq t }}^{2^n-1}|f((k+1)2^{-n}) - f(k2^{-n})|^{1/H}$$
converges uniformly to the linear function $\langle f\rangle_{t} = a_{H} t$, where $a_H$ is a constant depending only on $H$.
[1] Mishura, Yuliya, and Alexander Schied. "On (signed) Takagi–Landsberg functions: pth variation, maximum, and modulus of continuity." Journal of Mathematical Analysis and Applications 473.1 (2019): 258-272.