Does max current (amps) always flow through circuits?
The battery acts like a voltage source and so will try to provide a constant voltage (well, more or less) across the connected load.  The load will then draw whatever current it requires.  Open circuit?  Zero current.  LED with current limiting resistor?  I = (Vbat - Vled) / R.  

One thing to note is that a source can only drive a specific voltage OR a specific current.  The load will determine the other.  However, usually power supplies are rated something like 5V, 3A.  This usually means that it will supply a constant 5V at up to 3A.  The load could draw 0.1A, 1A, 2.5A, or even 0A from the supply.