Step 1: Define a Lewis acid.
A Lewis acid is a chemical species that can accept an electron pair.
Step 2: Analyze each species in the list.
\begin{itemize}
\item AlCl$_3$: Aluminum has an incomplete octet and can accept an electron pair. (Lewis acid)
\item NH$_3$: Nitrogen has a lone pair of electrons and can donate it. (Lewis base)
\item H$^+$: A proton has an empty 1s orbital and can accept an electron pair. (Lewis acid)
\item Co$^{3+$}: The cobalt(III) ion is electron deficient and can accept electron pairs, especially in forming coordination complexes. (Lewis acid)
\item $^-$OH: The hydroxide ion has lone pairs of electrons and can donate them. (Lewis base)
\item Mg$^{2+$}: The magnesium(II) ion is electron deficient and can accept electron pairs, particularly in aqueous solutions forming hydrated ions. (Lewis acid)
\item BF$_3$: Boron has an incomplete octet and can accept an electron pair. (Lewis acid)
\item Cl$^-$: The chloride ion has lone pairs of electrons and can donate them. (Lewis base)
\end{itemize}
Step 3: Count the number of Lewis acids.
The Lewis acids in the list are AlCl$_3$, H$^+$, Co$^{3+}$, Mg$^{2+}$, and BF$_3$. There are 5 Lewis acids.
Final Answer:
\[
\boxed{5}
\]