Lode Runner
Reaching level 150 in Lode Runner is possible; I never had the patience, nor time to leave the Apple //e running for the duration required. Thanks to a virtual machine, the game can be saved and resumed as necessary. Following level 150, the levels start over again, except the enemies are faster. This happens again at level 300.
Level 52 is tricky the second time through, and requires cutting corners to reach the falling enemy in time. The third time through, a different strategy is required, as the enemy is now too fast to reach in time. Instead, one must dig to get past the upper two enemies, then fall into the usual pattern.
Curtains fall at level 357, where the now twice sped enemy always intercepts one before crossing the first ladder (only eight bits were allocated for the level counter, as following level 256, it wraps, hence level 101 being displayed):
Note: lode_runner1.po on the Asimov Apple //e archive is corrupt at level 130-something. I am using lode_runner.do.gz. If paranoid, advance through the levels with control+U or control+6 to confirm the levels look right.
Next, Championship Lode Runner.