Why is the start symbol "not allowed" on the right hand side in Chomsky normal form?
Recall that in Chomsky normal form, we are allowed productions of three forms:


Productions of the form $A \to a$.
Productions of the form $A \to BC$.
The production $S \to \epsilon$.


We have to allow the production $S \to \epsilon$, since otherwise it will be impossible to generate the empty string.

Suppose for a moment that the production $S \to \epsilon$ is absent. Then we have the following very useful property:


  If $A \Rightarrow^* w$ then $w \neq \epsilon$.


This comes up in some proofs of the pumping lemma, for example.

If we have the rule $S \to \epsilon$, then we no longer have the property above. However, a similar property holds:


  If $A \Rightarrow^* w$ and $A \neq S$ then $w \neq \epsilon$.


This is due to our insistence of not having $S$ on the right-hand side of productions; otherwise we might have rules like $A \to SS$ and then derivations like $A \Rightarrow SS \Rightarrow^* \epsilon$.

If the rule $S \to \epsilon$ is absent, then there is no reason to disallow $S$ on the right-hand side of productions. Indeed, some authors allow $S$ on the right-hand side of productions as long as the production $S \to \epsilon$ is absent.