We don't know
El Duderino is not paid during the course of the film. He does "receive" a rug from the Big Lebowski (which is later repossessed by Maude), and I believe he keeps their clunky portable telephone, which ought to count for something. Maude also treats him to a thorough medical exam ("you will receive no bill"), which has non-zero economic value.
About that rug...
His Dudeness initially steals the rug by lying to Brandt, but Brandt explicitly forgives the theft when he calls later to recruit H.D. to be courier in the ransom scheme. Since he does this before H.D. agrees to participate, and without stipulating it as contingent on his participation, we must consider the rug a gift. And Maude later steals the rug back from him.
Importantly, it's a mistake to treat the stolen rug as fair compensation for Duder's original rug, because that rug was soiled by Jackie Treehorn's thugs, who had not been sent there by Mr Lebowski. Sadly, Duder's rationalization at the beginning of the film, that Mr L owes him because it's Bunny's lifestyle that resulted in the thugs' visit, does not hold water: it's Treehorn who owes him a rug (at the least). Treehorn never makes good on this debt, although I like to think that Treehorn's inexplicably urgent notepad sketch, of a man with a humongous penis, which Duder recovers through sheer investigative savvy, might become valuable some day.
About Maude's decent proposal...
Maude seems like a reasonably honest person and she can clearly afford the fee, but let's not forget that the Dude does not recover the ransom money, which is what she proposed to pay him for ("if you successfully do so, I will compensate you to the tune of..."), so I don't expect she would feel obligated to pay the 1%. That said, it's not out of the question that she might offer him some smaller quantity of bones or clams for his efforts. Nevertheless, the film ends without settling the matter.
As for the medical exam: it's thorough, and medical care in the U.S. is a valuable luxury, but it's not given as payment. She tells him it's recompense for the face-punch dealt to him by her friend (or bodyguard?), but the truth is that Maude has him checked out for the sake of her own health and the health of the child she intends him to impregnate her with. The situation is not far removed from Homer Simpson "gifting" Marge a bowling ball.
It's hard to evaluate the pregnancy, because Maude states he'll have nothing to do with the child, which I expect means he will be denied all the joys of parenthood except the brute fact of perpetuating his own genes (which is not exactly nothing). In any case, it's not payment.
About the Big Lebowski...
It seems clear Mr Lebowski intended from the beginning to stiff him, because he's a deadbeat loser the square community doesn't give a shit about, and that's exactly what he does. This is even less surprising considering that he is not exactly master of his own finances.
At the end of the film, the Stranger says things seem to have worked out pretty good for The Dude and Walter. I didn't find that to be the case, exactly. Jeff loses his friend Donnie, his car is significantly vandalized, and he's assaulted on multiple occasions including by a police officer. It's true that some good things happen to him as this stupefying story unfolds, but it seems a net negative (and in any case, none of those good things can be properly considered "payment").
The Dude doesn't get paid, and he almost certainly comes out a little behind.
Perhaps that's why it's significant that The Dude abides: although his life is materially, if marginally, worse, His Dudeness retains a seemingly positive attitude toward life.
