Z-transform of an affine function
That system cannot be described by a transfer function because it is not linear. It neither satisfies the condition of additivity nor does it satisfy the condition of homogeneity.
If $y_1[n]$ is the response to an input $x_1[n]$, and $y_2[n]$ is the response to an input $x_2[n]$, then the response to $x_1[n]+x_2[n]$ doesn't equal $y_1[n]+y_2[n]$ (additivity is violated). Furthermore, the response to $\alpha x_1[n]$ doesn't equal $\alpha y_1[n]$ (homogeneity is violated).
This is unrelated to existence questions concerning the $\mathcal{Z}$-transform. The $\mathcal{Z}$-transform of a constant $\beta$ doesn't equal $\beta$, but it simply doesn't exist.