Traveling salesman routing

All major OSM routing engines support this:

• Mapzen's Valhalla ("Valhalla also includes tools like time+distance matrix computation, isochrones, elevation sampling, map matching and tour optimization (Travelling Salesman)."),
• Mapbox's OSRM ("The trip plugin solves the Traveling Salesman Problem using a greedy heuristic...")
• Graphhopper uses the JSPrit library for route optimization ("TSP problem can be modelled by defining a vehicle routing problem...")

None of these services have a free and unlimited online offering (it would quickly be abused by people trying to save on their own AWS cost). Mapzen has an offer where you register a free API key and use that. OSRM doesn't need an API key, you can just use it. Graphhopper requires registration and while they have a free trial, I don't think they have a free tier.

All three are Open Source and you can install and use them without limits locally.