Chronic Pet Killer

A character whose pet, unfortunately, has died. Just like the last one. And the one before that, and the one before that...

Oddly, this trope can be played for laughs, fear, creepiness or tragedy depending on the work.

Examples of Chronic Pet Killer include:

Anime and Manga

In book two of RE:PLAY, a fictional interview with some of the lead characters reveals that lead singer Cree has killed four hamsters, a guinea pig and one unfortunate canary. Oh, and countless goldfish. Poor bastards...

In Harry Potter and the Methods of Rationality, Harry explains to McGonagall why he'd rather pay for his mail than buy an owl, or any familiar for that matter: "I had a pet rock once. It died." It's heavily implied later on that this actually happened. Maybe it was one of those "kid lab" "grow a crystal in a glass" sorts of rock, and he messed it up somehow?

Natalie Portman's character in Garden State.
The young Michael Meyers in Rob Zombie's Halloween goes through pet rats very quickly.
Darla in Finding Nemo.

Lenny from Of Mice and Men.
The Death of Rats in Discworld reaps the souls of a lot of mice, gerbils and hamsters that run into this. Susan has to replace the class pet at her school regularly.
- It's noted in Hogfather that the Death of Rats isn't very busy on Hogswatch Eve, but business picks up about three days later, when all the children who got mice, gerbils, or hamsters for Hogswatch start forgetting to feed their pets.
- In Wyrd Sisters it's revealed that Magrat has gone through quite a few familiars, invoking this trope.

Drusilla on Buffy the Vampire Slayer.
- Also Buffy "I killed my Giga Pet" Summers, herself. (This was part of an arc that drew parallels between these two protagonists left and right, so probably an afterthought rather than normal characterization.)
Cady in Reaper.
They're not exactly pets, but GOB from Arrested Development has terribly bad luck with animals used in his magic acts.
George and Reggie from Dead Like Me are noted to be like this.
Dwight Schrute from the American version of The Office.
There was a Saturday Night Live bit called "The Mellow Show" where the host would always forget to feed his pets...
Spencer from iCarly is known for "his many dead goldfish".

In the Lonestar video for the song Mr. Mom, the child chases the dog through the yard, which features a long line of pet tombstones.

In Dungeons & Dragons, Druids and Rangers have picked up this reputation among the player base. Since the only penalty for losing an animal companion is the ritual required to call a new one (unlike most other pets or auxiliary characters), using them as semi-disposable "meat shields" is a viable strategy.

Commander Shepard can become this in Mass Effect 2. Shepard's cabin aboard the Normandy features a fishtank, and Shepard can purchase fish from various vendors. The fish need to be fed between missions or they'll die, and have to be repurchased. Later on, Shepard's yeoman offers to take care of the fish, and from that point on it's no longer a worry.
Stephen Spookyraven's pets tending to be strangely short lived is the backstory explanation for why the Haunted Conservatory in the Spookyraven Manor in the game Kingdom of Loathing has unlimited numbers of small undead animals for you to kill.

Ariel of Drowtales has been Flanderized into this by the fandom even though she's only had three pets and one, the feral Melodia, is still alive, but either way she's still 1-2 on their survival rate. Fuzzy, the cat, was killed by her abusive older sister, while the other was a biogolem who she absorbed in order to escape someone trying to rape her (It Makes Sense in Context).
Prince Ricardo of No Rest For The Wicked is on his ninth horse named Artax.

Darla Sherman from Finding Nemo.
Elmira in Tiny Toon Adventures, though it's more chronic abuse than negligent killing...
Lisa went through about four cats in one episode of The Simpsons. The last one survives though.
Timmy from The Fairly OddParents was this, although he was told his pets just ran away. Really, all his animals died and were buried in the garden by his dad while Timmy was away at summer-camp... camp... camp...
- To be fair, nothing suggested that Timmy himself was that bad of a pet owner. His dad on the other hand...
In the Screwy Squirrel cartoon "Lonesome Lenny", a big dopey dog named Lenny keeps accidentally killing his companions. Whenever one of them would die, he would say, "You know, I had a little friend once, but he don't move no more". This is an unsubtle Shout-Out to Of Mice And Men, as mentioned above.
A variant in South Park, where it turns out that all the Paris Hilton's pets had committed suicide rather then putting up with their obnoxious owner.