big head, have a

Be conceited; have an exaggerated sense of one's own importance or ability. For example, The constant flattery of his subordinates is bound to give Thomas a big head. [c. 1800] Also see swelled head.

The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.