There are a couple of parts of the flight where the pilot is required to use the automation. The biggest is during cruise in what's known as RVSM airspace, where the vertical separation minimums are reduced from what was standard before RVSM was implemented. There, if your autopilot quits, ATC will send you down below the RVSM floor. RVSM is in use above some altitude in the 48 states and on transAtlantic routes. (I don't recall the exact altitude.)
The other is in flying an instrument approach to very low altitudes, known as a category III approach. IIRC, those must be flown on autopilot in order to continue below category III minimums.