Haven't thought about such trivial things in a long while. A cog railway goes right to the top of Mount Washington and train tracks appear less than a mile from both highpoints of Illinois & Iowa.
Taum Sauk Mtn seems to be the closest highpoint that Amtrak passes by with the train track about 3 miles away.
Any other train & hp trivia out there?
Not quite as close as the Amtrack rails to Taum Sauk, but you can ride Amtrack (Southwest Chief) out to Flagstaff to go climb Humphreys Peak. You can see Agassiz from the tracks (if you dodge the buildings in Flag) but Humphreys hides behind his shorter brother when you look up from the south.
When Nathan and I went out for the Konvention in '08, we rode Amtrack from Kirkwood, MO, via Kansas City out to Flagstaff. Definitely a relaxing way to get there if you can spend the time on the train. We had ridden the same route as far as Raton, NM, on our way out to Philmont in 2005, so we knew the routine and opted for the "low and slow" way as opposed to flying out.
Not a state highpoint, but one of the other well-known cog railways goes up Pikes Peak. That's a fun trip, too...