CARET is a 5 letter word that starts with C.

1. \Ca"ret\, n. [L. caret there is wanting, fr. carere to
   want.]
   A mark [^] used by writers and proof readers to indicate that
   something is interlined above, or inserted in the margin,
   which belongs in the place marked by the caret.
   
   

2. \Ca`ret"\, n. [F., a species of tortoise.] (Zo["o]l.)
   The hawkbill turtle. See {Hawkbill}.
   
   

^

Common: hat; control; uparrow; caret; itu-t: circumflex. Rare: chevron; intercal: shark (or shark-fin); to the ("to the power of"); fang; pointer (in Pascal).