I just tried out a free program, Translator's Abacus. You can drag and drop various file types (including PDF), and it pops up a browser with a printable report of the word count for each document. It worked fine for me. (It is specifically created for word counts and is only 435 KB... that is, not a "big application").
Alternatively: you can just Ctrl+A to select all text in Acrobat Reader and then copy-paste it into a program like Microsoft Word (which has a word count on the status bar at the bottom of the screen).