A character who hunts monsters for a living is surprisingly a pretty popular concept for games, anime, manga, or just comics in general. It can be used in a very simple way such as characters hunting monsters and making weapons/armor out of them on repeat (like in Monster Hunter) or as part of a very complex narrative (such as Attack on Titan).
The core concept still boils down to humans (mostly) hunting creatures way bigger than themselves and entirely based around hunting/killing monsters, whether it's for a career, for survival, or simply just for fun.
10 Inuyasha
Inuyasha was mostly seen as an action-romance rather than a monster-hunting anime. Though, with the recent release of Yashahime, maybe it's a perfect time.
Thinking back to the plot of Inuyasha, the growing bond between Kagome and Inuyasha was the foundation, but their purpose in working together in the first place was always to gather the remnants of the Shikon Jewel from all these different monsters/bandits. And, the original manga actually has a pretty nice style to it, where the black and white contrast adds to the designs for a lot of the monsters.
9 Toriko
Toriko is a fantastic example of the "hunting and cooking" sub-genre for series about hunting monsters, where the entire purpose for the cast of characters is to find new things to hunt so they can cook and create delicious new recipes. There's usually more to the overall plot than just that, especially in Toriko, but no matter what, cooking is always fundamental to the setting.
Toriko is no different and very likely played a large part in creating this sub-genre since the series has been out for over 12 years now. It's all about Toriko, of course, who is known as a Gourmet Hunter. This title he bears is saved only for the most expert of ingredient gatherers, with Toriko and the other Four Heavenly Kings at the peak.
8 Delicious In Dungeon
DiD is a seinen series that has been in publication since 2014 and puts a lot more focus on comedy and slice-of-life antics than it does on battling. And, to be fair, it is insanely funny. The camera is primarily focused on a party of four adventurers, but as with most series, the cast keeps growing over time.
This party's main goal is to get to the bottom of the dungeon in order to save Laois' sister and so they must eat all forms of different dungeon-dwelling lifeforms along the way to survive. It's obvious the author put a ton of time into thinking up these recipes and creatures.
7 Kaiju No. 8
Kaiju No. 8 is the more recent series on this list, but also one of the ones most focused on the act of actually hunting monsters or "Kaiju." Imagine if Attack on Titan got slammed together with Pacific Rim, and that's basically this series. In it, the world is constantly attacked by Kaiju of all sizes, and the Japanese Defense Force is the only group who seem capable of stopping them.
This "best of the best" crew was the end-goal for two childhood friends named Kafka and Mina, who had their home destroyed by monsters, yet Mina was the only one who made it there. Kafka, sadly, only ended up as part of the cleanup crew. But then, things take a bit of a turn. No spoilers on what exactly happens, but it's safe to say that there's a lot left to uncover in Kaiju No 8, and with the 19 chapters that are out so far, this series has a ton of potential.
6 So I'm A Spider, So What?
Much like most of the Isekai genre, So I'm a Spider, So What? is a manga that adapts a light novel of the same name. But, unlike other Isekai where the main character gets transported/reincarnated in another world with absurdly overpowered basic abilities, this one is about a human girl who reincarnates as a fragile little spider monster.
It's a monster-hunting series about a monster that hunts other monsters, growing, evolving, and changing along the way. It's one of the more absurd series out there and it does get ridiculous pretty quickly, similar to series like Re: Monster or Moon-Led Journey Across Another World.
5 D.Gray-Man
One of the older series on here, D.Gray-Man is another one of those classic Shonen franchises that people either completely adored or have never heard about. This one has some pretty religious overtones, with main character Allen Walker joining a group called the Black Order within the church and becoming an Exorcist.
What does he exorcise? Monsters called the Akuma mostly, created by some ancient evil being named the Millenium Earl. Unlike other monster-hunting series (such as Demon Slayer), D. Gray-Man lets the reader know early on that every Akuma used to be human and their deaths are a tragedy.
4 Chainsaw Man
Chainsaw Man is a high-octane and overly violent series. This story is about a dweeby and incredibly inexperienced teen named Denji, who gets lumped in with the Mafia from a very early age thanks to a mountain of debt his father left him.
A lot of stuff happens quite quickly after that, but the end result is that Denji fuses with his "dog", becomes a Chainsaw Devil hybrid, and joins the Public Safety Devil Hunters so that they'll provide for him. So, it's safe to say that this story gets wild pretty early on, and it only gets more absurd with every chapter.
3 Claymore
Claymore is a masterclass series that captures all the horror and violence of Berserk, the constant battles against tougher monsters like Solo Leveling, all while still telling quite the unique story about the badass female Claymore named Clare.
Clare is a Yoma-Human hybrid, which is exactly what qualifies her as a "Claymore." These are a group of man-made warriors by "The Organization" who are sent to fight Yoma all over the world. There's a lot more to Claymore, but to put it simply, this story is one of the all-time greats.
2 Solo Leveling
There are many good manwha out there, yet none of them even come close to Solo Leveling's popularity. With an anime on the way (likely in Summer 2021) and an ever-growing fanbase, Solo Leveling is a story entirely based around the act of hunting.
Like Hunter X Hunter, Solo Leveling is based entirely on a fictional representation of "Hunters", but these ones actually focus on hunting and killing monsters, unlike HxH which focuses on human-vs-human battles 90% of the time. In any case, Solo Leveling's power-scaling is so absurd that it's pretty much an Isekai series at this point, but it's still one of the best ones out there, with some truly stellar artwork to match.
1 Berserk
Berserk is the penultimate tragic seinen series about a tragic MC with an equally tragic life. It has horrifying monster designs, grotesque portrayals of how awful humanity can be, and a narrative that holds nothing back, no matter how difficult it is to look at. Berserk was started all the way back in 1990, and even though it's been running for all these years, the end is still a long way away.
Still, no matter how epic or longwinded the narrative is, at its core, it's all about Guts hunting monsters that are hunting him in order to survive. Nothing tops this series when it comes to complex settings, terrifying designs, or pure artistic detail, and it has influenced quite a huge number of series since it began.