a hamburger cooked with a slice of cheese on top of it.

1935–40, Americanism; cheese1 + -burger Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
cheeseburger (ˈtʃiːzˌbɜːɡə)
a hamburger cooked with a slice of cheese on top of it

Collins English Dictionary - Complete & Unabridged 10th Edition
2009 © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009
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Word Origin & History

1938, Amer.Eng., from cheese + ending abstracted from hamburger.
Online Etymology Dictionary, © 2010 Douglas Harper
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Example sentences
One such twist comes in cheeseburger spring rolls, a dish that sounds so
  misconceived you feel you have to try it.
But the technique will never generate a cheeseburger.
Some people would no sooner toast the season with eggnog than order a double
  cheeseburger, rare, with fries.
Traditional fare such as cheeseburger and fries are also on the menu.
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