The class of grammars recognizable by backtracking recursive-descent parsers


It is easy to show that there exists a grammar that can be parsed by a recursive-descent parser with backtracking but is not an $ \text{LL}(k)$ grammar (consider any grammar with a production featuring two alternatives starting with $ k$ occurrences of the same terminal).

My question is the following: Is there an identifiable strict superset of $ \bigcup_{k \in \mathbb{N}} \text{LL}(k)$ grammars that can be parsed by a backtracking recursive-descent parser, regardless of complexity?

If yes, is the maximal strict superset also identifiable?