Is there a largest class of halting programs?
There is no largest class of programs which have a decidable halting problem. If $L$ is such a class and $P \notin L$, then the class $L \cup \{P\}$ also has a decidable halting problem (why?). (This works since the class of all programs does not have a decidable halting problem.)

Responding to Raphael's comment, there is also no largest class of programs, up to finitely many exceptions, with a decidable halting problem. For each program $P$, let $L(P)$ be some infinite decidable set of programs equivalent to $P$ (for Turing machines, we can add dummy unreachable states, for example). Let $L$ be a class with decidable halting problem. I claim that there must be some $P$ such that $L(P) \cap L = \emptyset$. Otherwise, we could decide the halting problem for an arbitrary program $P$ by enumerating $L(P)$. Given a program $P$ such that $L(P) \cap L = \emptyset$, we can obtain a decider for $L \cup L(P)$, so that $L$ has infinitely many exceptions.