Step 1: Recall the conditions for the Cannizzaro reaction.
The Cannizzaro reaction is undergone by aldehydes that do not have any $\alpha$-hydrogen atoms in the presence of a strong base.
Step 2: Examine the structure of each given aldehyde.
\begin{itemize}
\item Phenyl ethanal (C$_6$H$_5$CH$_2$CHO): The $\alpha$-carbon (the carbon adjacent to the carbonyl group) has two hydrogen atoms. Thus, it has $\alpha$-hydrogens.
\item Methanal (HCHO): The carbonyl carbon is directly bonded to two hydrogen atoms. There is no $\alpha$-carbon, and hence no $\alpha$-hydrogens.
\item 2-Methoxy propanal (CH$_3$CH(OCH$_3$)CHO): The $\alpha$-carbon has one hydrogen atom. Thus, it has an $\alpha$-hydrogen.
\item Trichloro ethanal (CCl$_3$CHO): The $\alpha$-carbon is bonded to three chlorine atoms. Thus, it has no $\alpha$-hydrogens.
\end{itemize}
Step 3: Identify the aldehydes lacking $\alpha$-hydrogens.
Methanal and Trichloro ethanal are the aldehydes in the list that do not have $\alpha$-hydrogen atoms.
Step 4: Determine the number of aldehydes that undergo the Cannizzaro reaction.
Therefore, 2 aldehydes from the list will undergo the Cannizzaro reaction.
Final Answer: The final answer is $\boxed{2}$