Fearless Infant

We all know that babies cannot be killed. Many works, especially lighthearted ones, take this a step further and make it so that you can't even depict a baby being scared or disturbed by the scary and disturbing things happening around it. In fact, they'll often be depicted as enjoying whatever's happening, and they giggle and coo while a horrible monster is trying to eat them as they dangle over a pit of lava. Any Badly-Battered Babysitter in charge of these kids, of course, will be freaking out.

This is done because showing a baby screaming and crying in terror would be disturbing to the audience and instantly change the mood of the work, darkening it greatly. So showing the baby literally laughing at danger is a way of reassuring the audience "Don't worry, the baby's fine, we wouldn't go there." It should also be mentioned that babies have True Sight usually because of their innocence. They can see monsters' true forms. They just don't realize that they should be afraid. It's not stupidity so much as lack of life experience.

Examples of Fearless Infant include:

Anime

Yachiru from Bleach was revealed to be one of these in a flashback. (Now that she's grown to be an apparent eight years old, not only is she still fearless, she inspires fear in others)
Togepi in the Pokémon anime often went around oblivious to any danger, not even afraid of Team Rocket's Pokémon.
Melk the second from Toriko was found abandonned as a baby by Melk the first, a muscle bound shaggy giant with more scars than skin who could fit her in one hand. He was feared by all as the 'demon Gourmet', yet in two of the three panels showing the infant interact with the giant she was laughing in the embrace of a man who could squish her like a grape between his fingers.
- Apparently she could feel the demons 'love'.

Film

Baby's Day Out was pretty much built around this trope. The baby’s crawling all over the city with frantic caretakers and bad guys chasing him and taking the Amusing Injuries.
The baby in Raising Arizona.
Pubert Addams in Addams Family Values has a number of things happen to him that would terrify a normal baby while he laughs and coos. Justified in this instance, though, because he's an Addams.

Literature

Exaggerated example: Baby Sunny from A Series of Unfortunate Events and The Movie based on the first three books of the series.
Harry Potter had literally just witnessed the death of his parents and escaped death at the hands of Voldemort, and was then flown by motorcycle to the doorstep of the Dursleys. We don't hear much of him crying in the book, and in The Movie he is more or less dozing off.
The youngest Carpenter baby in The Dresden Files has an "ooh shiny!" reaction to one of the Denarian coins; something Harry knows is extremely dangerous.
The Jungle Book: When baby Mowgli first encounters the leaping Father Wolf he just laughs. He then reacts calmly to the wolf picking him up by the scruff of the neck and taking him to Mother Wolf, and quickly pushes her other cubs aside so he can get at her milk. He doesn't even seem too flustered when Shere Khan tries to force his way into the wolves' den. Later on when the wolf pack are deciding whether he should live or die, he just plays with some pebbles.

Mythology

Older Than Feudalism: Baby Hercules's first act of heroism was to grab the two snakes sent into his bed and wrestle them to death while his mortal half-brother cowered away.

Western Animation

In The Incredibles, the entire family is having a loud, angry argument around the dinner table, and baby Jack Jack simply coos and laughs. Averted, however, when Syndrome kidnaps Jack Jack, and the baby cries and screams, before becoming angry and displaying his superpowers.
In Megamind, Metroman is showing off for an audience, and picks up and starts juggling babies. Instead of being terrified, they all just giggle.
WALL-E: When the Axiom is listing, causing everyone to slide along the tilted floor, a group of babies are among those sliding. They're only heard crying for about a second before being caught, whereupon they immediately launch into cooing and giggling, even through the adults are still screaming.
In Tarzan, Baby Tarzan is pretty calm through the ship sinking, and later giggles and coos while being thrown around in the battle as Kala desperately tries to defend him (and herself) against Sabor.
Boo's reaction to the monsters in Monsters, Inc..
The character Baby Herman plays at the beginning of Who Framed Roger Rabbit?? It's not actually Baby Herman using this trope, though, as he's only an actor.
Baby Kal-El doesn't mind being bundled into a rocketship and fired into space.
- By the same token, neither do the babies who grow up to be Metro Man and Megamind.
Swee'Pea from Popeye would happily crawl over girders high above the city.
Mindy from Animaniacs would lead Buttons into multiple dangerous situations, blithe as you please.
Bamm-Bamm from The Flintstones: Justified example, though; nothing scares a baby who could walk from birth and has Super Strength.

Real Life

The Lioness wants to eat the baby video. The baby cried when taken away from the glass.
Truth in Television to a degree. Fear of heights, drowning, etc. does not kick in instantly after birth. Some individuals are crazy enough to not fear at all and become daredevils and the like later in life.