Passing over the great Shakespeare debate, I’ll throw in Pushkin for a pushpin.
Incidentally, a lot of people I’ve talked to say he’s the Russian Shakespeare.
I don’t understand it myself, but that may be to do with not being a native speaker of Russian.
I’m fond of O’Henry. Of course, after a while he was unable to top himself and the irony twists got piled on too thick. But his early stuff will stand the test of time.
I think assertions that Shakespeare is unarguably the best writer (were such assertions forthcoming, which i’m not quite sure they are) render that acclamation worthless - ‘best’ is an expression of measurement - if measurement is forbidden, ‘best’ cannot be soundly established.
From an American perspective, I’ve always championed William Faulkner. I think his name should stand right along with the rest of those mentioned.
Also, James Joyce?
For German language writers, Goethe is the towering giant.
Shakespeare, of course, is a close second. 
No love for Papa? And, before you scoff, let me say that, in my life, I have gone from an outright Hemingway-hater to a complete devotee. I don’t think there’s any debate that he was immensely influential. He at least deserves a look.
I don’t think that we can discount Shakespeare because he’s too specialized. There are probably very few (or maybe no) writers who have created works of literature in every genre. Plus, prose as the ultimate in literature is a really modern view.
From what I’ve gathered after taking a few Russian literature courses, Pushkin and Tolstoy are fiercely revered.
For American writers, I’d say Twain, Hawthorne, and Hemingway (none of whom I particularly like). Judging American writers is a bit easier because there’s less history there; one doesn’t have such an anchored base for comparison as Shakespeare.
Sebhal asked about the greatest female writer. In English, I’d offer George Eliot.
I was trying to remember if there is a Korean writer that has Shakespeare’s status in Korean culture, but I can’t think of one. Hm.
How does one pick the greatest writer anyway? There are so many different standards we can use to judge writers by. In terms of sheer influence there are few that are going to top Shakespeare: James Joyce and T.S. Eliot are two that come to mind as possibilites. I think both of them were deliberately jerking a lot of strings in the literary community though (both of them admitted as much, I think).
But Dickens was a novelist and short story-writer, and in those forms his genius is certainly undeniable, however he wrote serials and streteched his plots to fill the serial publishing requirements, and he has no real poetry to evaluate, it’s all prose and narrative. So he can’t really be compared to Shakespeare.
In other words, if you focus too hard on the type of writing being done, there is no answer to the OP, because no one is a master of poetry, prose, plays, novels, short stories, narrative non-fiction, etc.
I don’t think anyone is suggesting you can’t argue that he wasn’t the greatest writer. But it seems some people are arguing that he wasn’t a writer at all, which is a pretty bizarre claim. I mean, c’mon! “Playwright sounds like shipwright?” Someone actually put this argument forward with a straight face?