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[kryt3n]

Runaway is labeled as a "test version" (i.e. demo), a side-view adventure game in which you control a monkey. You escape the cage by plugging up a "sink" (with the banana) so that the zookeeper comes by and you can swipe his key to the cage. Other than your recollection of EGA graphics, the game elements match your description so closely that if Runaway is not exactly the game you remember then perhaps it is a remake with VGA graphics. In any case, I wish you good luck in finding the game you seek.