  What limits the value of the superconducting transition temperature ($T_c$)
is a question of great fundamental and practical importance. Various heuristic
upper bounds on $T_c$ have been proposed, expressed as fractions of the Fermi
temperature, $T_F$, the zero-temperature superfluid stiffness, $\rho_s(0)$, or
a characteristic Debye frequency, $\omega_0$. We show that while these bounds
are physically motivated and are certainly useful in many relevant situations,
none of them serve as a fundamental bound on $T_c$. To demonstrate this, we
provide explicit models where $T_c/T_F$ (with an appropriately defined $T_F$),
$T_c/\rho_s(0)$, and $T_c/\omega_0$ are unbounded.
