We consider which ordinals, with the order topology, can be Stone-\v{C}ech remainders of which spaces of the form $\psi(\kappa,\mathcal M)$, where $\omega\leq\kappa$ is a cardinal number and $\mathcal M\subset[\kappa]^\omega$ is a maximal almost disjoint family of countable subsets of $\kappa$ (MADF). The cardinality of the continuum, denoted $\mathfrak c$, and its successor cardinal, $\mathfrak c^+$, play important roles. We show that if $\kappa>\mathfrak c^+$, then no $\psi(\kappa,\mathcal M)$ has any ordinal as a Stone-\v{C}ech remainder. If $\kappa\leq\mathfrak c$ then for every ordinal $\delta<\kappa^+$ there exists $\mathcal M_\delta\subset[\kappa]^\omega$, a MADF, such that $\beta\psi(\kappa,\mathcal M_\delta)\setminus\psi(\kappa,\mathcal M_\delta)$ is homeomorphic to $\delta+1$. For $\kappa=\mathfrak c^+$, $\beta\psi(\kappa,\mathcal M_\delta)\setminus\psi(\kappa,\mathcal M_\delta)$ is homeomorphic to $\delta+1$ if and only if $\mathfrak c^+\leq\delta<\mathfrak c^+\cdot\omega$ (ordinal arithmetic).