Is there a word that describes a person who constantly exaggerates?
Exaggerator and overstater are both English words that fit your meaning.
How about histrionic? Or drama queen?
How about braggart? ~ " A vain bragger, one who brags much."
http://www.oed.com/view/Entry/22483?redirectedFrom=braggart#eid
Bullshitter comes to mind, although it is potentially offensive.