Sentinel 2 imagery as a webservice in a leaflet or openlayers map

The closest you can get to that are EOX's cloudless map tiles: https://s2maps.eu/

These are not latest, but they are global, cloudless and they are free (CC BY 4.0). WMS and WMTS options available.

The problem with "the latest" imagery is that it will often be useless due to clouds. This makes it difficult to justify the processing and storage costs as mentioned in one of the comments above. EOX was processing the data for 2 weeks on 20 servers. One should theoretically do something similar on a weekly basis (a bit less computing as there is no need to select the best pixel). This is why we (disclaimer: I am coming from the above mentioned "sentinel hub" company; we apologize for the confusion with the name, it was not intentional) rather opted for the approach when the user can choose the configuration of her likeness). But this comes with some costs as well...

That being said, ESA and Copernicus have initiated a DIAS project, which should offer free WMS services as well, so you might find what you are looking for in spring 2018.