I See a River is actually Urinetown's finale number, followed by a short epilogue. An 11 o'clock number is a big powerhouse ballad just before the end of the show, usually performed by the lead (but sometimes can be a duet) to energize the audience before reaching the finale. They are most often found in the old-fashioned musical comedies, but sometimes in the dramas as well, but they are not in every musical and they are not always the most popular song.
Some examples:
Cats - Memory
Company - Being Alive
Mamma Mia - The Winner Takes it All
Cabaret - Cabaret
A Little Night Music - Send in the Clowns
The Sound of Music - Edelweiss (thought I was going to say Climb Every Mountain, didn't ya?
South Pacific - This Nearly Was Mine
Zorba - I am Free
Phantom of the Opera - Past the Point of No Return
West Side Story - I Have a Love
Chicago - Nowadays
"What can you expect from a bunch of seitan worshippers?" - Reginald Tresilian