HAZENS, N.H.—Some people collect stamps, others collect bird sightings and a few count the state capitol buildings they visit. Mike Rose is riding a train to this remote railroad junction in the White Mountains of New Hampshire to track railroad miles.

Mr. Rose's journey takes him past steep cliffs, towering trestles and cascading brooks. Vitally important is that he is riding over this track for the first time and can therefore add it to his collection. "I got the mileage—that's all that counts," says Mr. Rose, a 65-year-old...