Lipo charge has 2 parts - the 1st is constant current (the 5amp you thought it would stay at). Then when max cell voltage of 4.2v is reached, the 2nd part of the charge begins - constant voltage. Charger holds the voltage constant and you'll see current slowly fall off as the batt tops off. This phase typically takes the longest time and puts in the smallest part of the charge, somewhere around 10%. You'll see this referred to as "CC/CV" lipo charge routine.

Chargers are beginning to feature a lipo fast charge option that does away with the CV part of the charge and you run with a +/-90% charge to save time.