The demand elasticity for most goods increases the longer the time frame that is considered. For example, in the short term, the demand for gasoline is very inelastic. If the price of gasoline triples in the next week, people will probably keep buying at near the same rate. In the long run, the demand for gasoline is much more elastic. If the price of gasoline stays at that price for a decade, people will learn to use more mass transportation, buy more fuel efficient cars and live closer to work in order to limit their need for gasoline.
Some goods, however, follow the opposite time trend. Consider your demand for air, if someone flatulates in your vicinity, the cost of breathing increases significantly. In the short run, your demand for air is quite elastic, because you can hold your breath. In the long run, your demand for air is much more inelastic. If the air stays stinky, you still have to breathe.
While demands which are less elastic in the long run than in the short run exist, I can’t think of any markets where this might have important implications.