That's an interesting question indeed~ Tbh though, I have a hard time thinking of examples since most selfish characters I've come across are written a bit ham-fistedly, and so I don't tend to enjoy nor care to remember them.
However, I do have one character that comes to mind. I don't really know if other people would agree that he is well-written, but Kei Nagai from Ajin is my choice for one of the most selfish characters that I nevertheless adore. Though, it might be more accurate to say that he is sociopathic rather than selfish, but the main concept is still relevant.
What makes Kei tolerable for me as a protagonist is that his selfishness is logic based. He will always put himself first, because humans are naturally self-preserving, and he'll exploit other people's generosity if it serves to benefit him because why not? It's not like he forced them to do so. But that's all - he's only interested in bettering himself. He won't go out of his way to interfere with others if they don't bother him, and if what he does ends up benefiting someone else, whatever.
It's a cut-throat ideology, but it makes sense when you consider the whole of his character as well as the nature of his supporting cast. The most important part of making a flawed character likeable is to reasonably justify their actions. It's also muchhh easier if you don't let your characters become sadistic. It's hard to sympathise with someone who enjoys other's pain for the sake of it.
So, basically, if you follow @shazzbaa's guideline, Kei falls into the 'Competent' type. All the reasons she lists at the end of her post are essentially why I love Kei and root for him, even though I think he's a pretty shitty person. XD
Hope that gives you some insight, and good luck with your writing~!
(Also, on a side note: I also absolutely adore Kei's 'best friend' and polar opposite, Kaito, who is much more of a stereotypically likeable character. However, I actually think I like him so much - even more than I love Kei, really - because of his relationship with Kei. Due to Kei's absolutely shite personality, it makes me feel that much more for Kaito's compassion. I'm not sure if this is relevant to the main topic, but I just thought it was interesting.)