There are incredibly many such flights; if you want a list of AA's flights, download their PDF timetable and find all the flights which list "1" in the "number of stops" column. You may be able to do something similar with other carriers' timetables.

As another example, go to the "From Hong Kong" section of the CX timetable; find all the "1 stop" flights: in this case, you can get a one-stop flight from HKG to CNS, CMB, DXB, FUK, KHI, MEL, BOM, NGO, JFK, KIX, PEN, RUH, ICN, SIN, NRT, or YYZ on CX. (DXB has a 2-stop flight as well.)

The big drawback to these flights is that you only get miles for the equivalent nonstop routing: e.g., if I fly SJC-DFW-AUS on separate flights, I get around 1938 miles because the DFW-AUS leg counts for 500 miles. But if it's a single flight number, it counts the same as a nonstop SJC-AUS, or only 1476 miles. Thus, I'd much rather book two different flights if the price is the same, unless there's a concern about delays causing a missed connection. (Missed connections are quite rare with a single flight number!)