The Editor holds 12 systems of 16 cells, the twelve lines of chords counting down the page, which is 192 cells in all. New songs can start from a barline template of up to 96 measures.
 
When a chart runs out of room, make the bars narrower rather than looking for another line. Delete the spare cells in the bars above and the space appears below, fitting eight or more bars on a line. A measure holding a single chord can be squeezed to one cell, with a barline on each side, and the small chord font gives back more room still.
 
The grid does not warn you when it is full. Once the chart reaches the last cell, anything pushed past it is dropped without a message, so check the end of your chart after inserting space near the beginning.
 
The Bossa 3 exercise included with iReal Pro shows how compact a chart can get.
 
For more, see Spacing chords and bars on the iReal Pro website.