Reachability queries checking the existence of a path from a source node to a target node are fundamental operators for querying and processing graph data. Current approaches for index-based evaluation of reachability queries either focus on plain reachability or constraint-based reachability with alternation only. In this paper, for the first time we study the problem of index-based processing for recursive label-concatenated reachability queries, referred to as RLC queries. These queries check the existence of a path that can satisfy the constraint defined by a concatenation of at most k edge labels under the Kleene plus. Many practical graph database and network analysis applications exhibit RLC queries. However, their evaluation remains prohibitive in current graph database engines. We introduce the RLC index, the first reachability index to efficiently process RLC queries. The RLC index checks whether the source vertex can reach an intermediate vertex that can also reach the target vertex under a recursive label-concatenated constraint. We propose an indexing algorithm to build the RLC index, which guarantees the soundness and the completeness of query execution and avoids recording redundant index entries. Comprehensive experiments on real-world graphs show that the RLC index can significantly reduce both the offline processing cost and the memory overhead of transitive closure while improving query processing up to six orders of magnitude over online traversals. Finally, our open-source implementation of the RLC index significantly outperforms current mainstream graph engines for evaluating RLC queries.