(Don't worry, I'm just doing this so I have a little backlog for my LP... unless someone requests a monster!)
Anyway, the Undying One. My first Undertale fan-dude. An apologetic observer who has great power since he's made of a diety. However, being put inside 'the box' to protect it... integrated him, making him lose much of his ability to actively stop the player. Yes, he kills and resurrects you once, but on request; he's too slow otherwise.
In fact, it's implied other people (or Ophinia, on a genocide run and after meeting her) carry him around.
Despite his name, he is NOT immortal. He's a rock, and monsters have trouble breaking rocks. When first met, he sneaks into Toriel's house at the smell of cake (on the back of a dog) and is entirely unphased by her fire magic. His immunity to magic and monster mumbo means the entire underground just assumes he's immortal. However erosion, pick axes and steep drops can still break him (though he still lives, but in less useful sizes).
He never fights the player directly, unless asked to do so. He is only REALLY killed by kicking him off ledges, uses items and whatever, and can only be killed on a genocide run as the prerequisites do not occur on neutral runs. Despite being kind and forgiving, he never gives the player anything but tips, as he has nothing to hold or give with.
Interestingly, he has a subplot that is resolved by a Toby-Dog walking in, claiming it's not cannon' and walking off. This involves the Evil Presence, a powerful being that is the Fallen Child, though it's quickly implied that it cannot be killed and must be fought with kindness. He makes it visible and fightable, and upon a spare, the dog claims this never happened and its done. Nothing for your troubles.
He appears before the Ultra Pacifist Boss, insisting you be gentle with a chaotic soul, furthering his unwillingness to admit violence as a means to solve problems.
He can be challenged to a fight, but you REALLY gotta push him. Maybe get him some flowers to look at too.
The evil presence has attacks too!