In all of political comedy, there may be one person who truly understands Donald J. Trump. And that person is a dog.

In “Triumph’s Election Special 2016” on Hulu, Triumph the Insult Comic Dog — the cigar-chomping, foul-muzzled canine puppet voiced by comedian Robert Smigel — offers to coach the Republican presidential candidate Mike Huckabee on belittling his opponents in his next debate. (The special was taped; Huckabee has since dropped out of the race.) “I’m an insult comic,” Triumph explains. “That’s the wave of the future. That’s what Trump is doing.”

It takes one to know one. (You can’t, after all, spell “Triumph” without “Trump.”) In a few quick sentences, Smigel both summed up Trump’s political style and explained why TV comedians have had a hard time satirizing him in any memorable way. How do you spoof a candidate who treats campaigning like a roast?

Trump is now a serious candidate — often a self-serious, angry one — with a serious chance. But stylistically, he works in the mode and rhythms of a stand-up. He riffs. He goads. He works blue. When he gave a victory speech in New Hampshire, feinted at congratulating his opponents, then pivoted — “Now that I’ve got that over with …” — he sounded like a sketch comic doing an imitation of himself.

His style has rendered him, weirdly, almost comedy-proof. Election parodies traditionally exaggerate candidates. But Trump exaggerates himself — he’s the frilled lizard of politics, inflating his self-presentation to appear ever larger. Satire exposes candidates’ contradictions and absurdities. But Trump blows past those, while his supporters cheer.

Whatever anyone thought of Trump as a candidate, the consensus was that he would be a one-man stimulus program for comedians. He’s given them plenty of material, but little of it has stuck. As Trump has defied conventional politics and confounded conventional pundits, so has he frustrated conventional satire.

There were a few successes early in the campaign. The new “Daily Show” host, Trevor Noah, a native of South Africa, used his own background to find a novel angle on Trump, likening him to an African dictator a la Idi Amin, given his “lavish lifestyle” and “level of self-regard.”

But more often comics have been scrambling to keep pace with the news. A December “Saturday Night Live” sketch tried to exaggerate Trump’s debate attacks on Jeb Bush: “I know for a fact that you pee sitting down.” Transgressive, right? By February, Trump was repeating a rally participant’s obscene insult of Ted Cruz.

Nonpartisan major-network comedy is used to mocking the foibles of candidates who accept certain standards of public presentation — Marco Rubio, say, going on auto-repeat in New Hampshire — but Trump doesn’t much care about those. The more opinionated cable shows, in the mold of Jon Stewart’s “Daily Show,” like to point out politicians’ dog whistles, but Trump uses an air horn.

So, with Trump and his movement, a comic can often only point and wince. On “The Nightly Show” in January, Larry Wilmore noted that, years after Trump’s charge that Barack Obama was not born in the United States was debunked, many of Trump’s followers didn’t care; 20 percent of Americans still believed it. “I don’t have a joke for this,” he said. “I just want that [expletive] to sink in.”

TV’s comics struggled with another new candidate in 2008: Obama, whose historic potential to become the first African-American president may have left them skittish. The most effective Obama spoof of 2008 on “SNL” was a meta-joke about someone else’s inability to be tough on him: a debate sketch had moderators grilling Hillary Clinton while fawning over Obama. (“Are you comfortable? Is there anything we can get for you?”)

With Trump, on the other hand, it’s tempting to spoof his superficial attributes (the hair, the accents) over the dark, polarizing stuff (the plan to ban Muslims, the supporters attacking protesters at his rallies).

Stewart, a veteran of the Trump rodeo, might have been able to balance mocking Trump while treating him as a real, even serious phenomenon. Seth Meyers of “Late Night” occasionally manages that, as when he recently slammed the notion that Trump and Bernie Sanders are two sides of the same coin: “Sanders has run a serious campaign on issues. Trump is a demagogue, running entirely on personality. … The only things they have in common are they are bad at using combs and they both pronounce it ‘yuge.’”

Many late-night hosts this year are general-election rookies: Noah, Wilmore — even Stephen Colbert is new at playing himself on CBS’ “Late Show” while moderating his act for a broad audience. TBS’ Samantha Bee has shown early promise, but John Oliver’s “Last Week Tonight” is purposely not fixating on the election cycle. And while the get-along guy Jimmy Fallon has a technically fine impression of Trump, it’s all hair and no teeth.

Two of the best takes on Trump came last week from outside conventional TV. “Funny or Die Presents Donald Trump’s The Art of the Deal: The Movie,” released online Wednesday, purports to be a never-aired 1988 TV movie, directed by and starring Trump (Johnny Depp). Depp, in fleshy makeup that makes him look like a turtle wearing a tuft of cotton candy, plays Trump as a vain tyrant, berating his soon-to-be-ex-wife Ivana (Michaela Watkins) and obsessing over his image.

The movie skewers Trump’s current rhetoric retroactively; he praises the Taj Mahal as “easily the classiest thing ever built by a Muslim.” But it also targets his ego, presenting him in washed-out VHS video as a rerun, a period piece, an ’80s relic with crippling insecurity. (One jab that really got under his skin, after all, was when the humor magazine Spy labeled him a “short-fingered vulgarian” in the 1980s.) The meta-conceit, that Trump made the film as a vanity project, means that everyone else in it is the product of his self-flattering imagination; the world, as its protagonist sees it, is a movie in which he is the only character.

Of course, to a man of Trump’s healthy self-esteem, being played by one of Hollywood’s biggest stars is simply confirmation of his hugeness. “Triumph’s Election Special” (also from Funny or Die), which followed several campaigns on the road in Iowa and New Hampshire, unloads some conventional zingers at Trump. (“He’s written 15 books and they all start on Chapter 11!”) But its most blistering takedowns focus on his supporters, the populist mosh pit that has kept him from hitting the floor despite his verbal stage dives.

In one segment, Smigel/Triumph hires “multiple robotic white women” to pose as Fox News reporters, asking voters to react to fictional Trump scandals, including a proposal to “sterilize Puerto Ricans until we ‘figure this thing out.’” “He’s just saying what everybody else believes,” says one supporter. Another, asked what he would say to to people criticizing Trump for delivering “a three-minute monologue in an insulting Chinese accent,” blows a raspberry into the microphone.

The “Borat”-style stunt is funny because it’s true — or rather, because it tests the limits of what Trump’s fans believe could be true and what they’d defend anyway. Most important, the bit, however gonzo, is about the content of Trump’s campaign: the harshness of his proposals and the passions he’s appealing to.

(END OPTIONAL TRIM.)

Some comedy critics might call this “punching down,” that is, picking on regular people rather than the powerful. But to ignore the role, and the collective power, of the millions of people helping Trump upend traditional politics is to ignore why his candidacy is news (which is to say, worthy of satire) in the first place. If Triumph shares one belief with Trump, it’s this: Sometimes insults are another way of showing that you care.

© 2016 The New York Times Company