$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of all functions of the form $\sum_{i=1}^n c_i f_i$, where $n$ is a natural number, the $c_i$'s are nonnegative real numbers, and the $f_i$'s are in $F$.
Is it true that every convex function from $\R^2$ to $\R$ is the pointwise limit of a sequence of functions in $G$?
The Laplacian measure of a function in the closure of $G$ is the weak limit of linear combinations of the Laplacian measures of your elementary functions, whose support are "tripods", that is, unions of three rays emanating from a vertex.
In particular, the support of the Laplacian measure of a function in the closure of $G$ must be a union of tripods. Take a maximum of three independent linear functions that is lower bounded, and take the max of that and a positive number. This function has a Laplacian whose support cannot be expressed as a union of tripods.
The answer is negative. Example: $$\max\{ |y|,|x|-1\}.$$ This is a convex piecewise linear function which is not a convex combination of your "tripods", and not a limit of such combinations.
