The largest sum of coin currency one can have without being able to give change for a dollar is $1.19. This is made up of three quarters (or a single quarter and a single half-dollar coin), four dimes and four pennies.
The problem is solved by considering first the largest denomination of coin and working down to the smallest. A half dollar gives 50 cents to start. Two half dollars would give change for a dollar, but adding a quarter would give 75 cents. Another quarter added to the mix would be enough for a dollar. Three dimes is just over a dollar, yet cannot provide exact change. Four dimes also cannot give exact change, but five dimes can, so the maximum number of dimes must be four. A nickel added to the collection can provide change enough for a dollar by pairing the nickel with two of the dimes. Since nickels cannot be used, pennies, the lowest denomination, must now be considered. Since five pennies would be enough to make change, the largest number of pennies is four, adding up to a total of $1.19.