  For two-dimensional percolation on a domain with the topology of a disc, we
introduce a nested-path operator (NP) and thus a continuous family of one-point
functions $W_k \equiv \langle \mathcal{R} \cdot k^\ell \rangle $, where $\ell$
is the number of independent nested closed paths surrounding the center, $k$ is
a path fugacity, and $\mathcal{R}$ projects on configurations having a cluster
connecting the center to the boundary. At criticality, we observe a power-law
scaling $W_k \sim L^{X_{\rm NP}}$, with $L$ the linear system size, and we
determine the exponent $X_{\rm NP}$ as a function of $k$. On the basis of our
numerical results, we conjecture an analytical formula, $X_{\rm NP} (k) =
\frac{3}{4}\phi^2 -\frac{5}{48}\phi^2/ (\phi^2-\frac{2}{3})$ where $k = 2
\cos(\pi \phi)$, which reproduces the exact results for $k=0,1$ and agrees with
the high-precision estimate of $X_{\rm NP}$ for other $k$ values. In addition,
we observe that $W_2(L)=1$ for site percolation on the triangular lattice with
any size $L$, and we prove this identity for all self-matching lattices.
