Is there a web service showing me the LEGO set with the most parts?
Brickset has a data mining page which sorts sets by certain factors including new sets, unreleased sets, most minifigures in a set, best/worst price ratio and best British pound to American dollar exchange rates.

The sets with the most pieces can be found here. As of December 2018, the set with the most parts is the Star Wars 75192 Millennium Falcon, with 7541 pieces.