Motivated by many practical applications in logistics and mobility-as-a-service, we study the top-k optimal sequenced routes (KOSR) querying on large, general graphs where the edge weights may not satisfy the triangle inequality, e.g., road network graphs with travel times as edge weights. The KOSR querying strives to find the top-k optimal routes (i.e., with the top-k minimal total costs) from a given source to a given destination, which must visit a number of vertices with specific vertex categories (e.g., gas stations, restaurants, and shopping malls) in a particular order (e.g., visiting gas stations before restaurants and then shopping malls). To efficiently find the top-k optimal sequenced routes, we propose two algorithms PruningKOSR and StarKOSR. In PruningKOSR, we define a dominance relationship between two partially-explored routes. The partially-explored routes that can be dominated by other partially-explored routes are postponed being extended, which leads to a smaller searching space and thus improves efficiency. In StarKOSR, we further improve the efficiency by extending routes in an A* manner. With the help of a judiciously designed heuristic estimation that works for general graphs, the cost of partially explored routes to the destination can be estimated such that the qualified complete routes can be found early. In addition, we demonstrate the high extensibility of the proposed algorithms by incorporating Hop Labeling, an effective label indexing technique for shortest path queries, to further improve efficiency. Extensive experiments on multiple real-world graphs demonstrate that the proposed methods significantly outperform the baseline method. Furthermore, when k=1, StarKOSR also outperforms the state-of-the-art method for the optimal sequenced route queries.