When something is inexhaustible, you'll never run out of it. If your grandfather appears to have an inexhaustible supply of nostalgic stories, it seems like he could tell them forever.
Unfortunately, if the kids you're babysitting have inexhaustible amounts of energy, they are never going to fall asleep. But if you've got an inexhaustible supply of Disney movies for them to watch, you'll be able to keep them distracted until their parents come home. This adjective combines the prefix in-, "opposite of," and exhaustible, from the Latin exhaurire, "take away, use up, or empty." So when something can't be used up, it's inexhaustible.