There is no definitive answer to this question as it is subjective. However, based on the available evidence, we can narrow it down to a few contenders.
One contender for the title of smallest mouth belongs to the pygmy marmoset, a species of monkey that is native to South America. These monkeys are incredibly small, with an average body length of just over 5 inches (13 cm). Their mouths are also proportionately tiny, and they have been known to close them completely when they sleep.
Another possible candidate for smallest mouth honors goes to the bar-headed goose. This bird is native to Asia and is one of the highest-flying birds in the world, capable of reaching altitudes of over 29,000 feet (8,800 m). They have very small heads and beaks in comparison to their body size, and their mouths are correspondingly minuscule.
And finally, we come to the elephant shrew. This creature gets its name from its long snout which resembles that of an elephant’s trunk. It also has a very small mouth in relation to its overall size. Interestingly enough, despite having such a tiny mouth, elephant shrews are able eat up twice their own weight in food every day!
So there you have it: three contenders for the title of animal with the smallest mouth. In terms of sheer size alone, it seems like the pygmy marmoset has got this one sewn up; but when you take into account how much these creatures can eat relative to their body size (and therefore how big their mouths must be), then perhaps the elephant shrew deserves top spot after all!