  We study nodal component count of the following Gaussian Laplace
eigenfunctions: monochromatic random waves (MRW) on $\mathbb{R}^2$, arithmetic
random waves (ARW) on $\mathbb{T}^2$ and random spherical harmonics (RSH) on
$\mathbb{S}^2$. Exponential concentration for nodal component count of RSH on
$\mathbb{S}^2$ and ARW on $\mathbb{T}^2$ were established by Nazarov-Sodin and
Rozenshein respectively. We prove exponential concentration for nodal component
count in the following three cases: MRW on growing Euclidean balls in
$\mathbb{R}^2$; RSH and ARW on geodesic balls, in $\mathbb{S}^2$ and
$\mathbb{T}^2$ respectively, whose radius is slightly larger than the
wavelength scale.
