Jim Newell, "Is Donald Trump’s Favorite Term Bigly or Big League? You Make the Call", Slate 9/24/2015:
What is that word—or words—that Donald Trump throws into the middle of basically everything he says?
The consensus around the LLOG water cooler was "big league", but I don't think we ever wrote about it. The Federal News Service transcript of last night's verbal brawl agrees:
I will say this. Mitt Romney looked like a fool when he delayed and delayed and delayed. And Harry Reid baited him so beautifully. And Mitt Romney didn’t file his return until a September 21st of 2012, about a month-and-a-half before the election. And it cost him big league.
But lots of people are convinced it's "bigly":
Update: he said "bigly" https://t.co/LXAllFhejx
— Silvia Killingsworth (@silviakillings) February 26, 2016
A sample of other Trump "big league"/"bigly" performances is here:
I continue to hear all of these as "big league", though it's also possible to hear (some of) them as "bigly" given the strong initial-syllable stress, the (sometimes) unreleased final [g], and the short second syllable.
And in fact, bigly has a fine pedigree, according to the OED. In the sense "With great force; firmly, violently; (also) stoutly, strongly", it goes back to the 15th century:
c1400 (?c1380) Patience l. 321 Þe barrez of vche a bonk ful bigly me haldes.
a1470 Malory Morte Darthur (Winch. Coll. 13) (1990) I. 416 So rowghly and so bygly that there was none myght withstonde hym.
And a sense glossed as "Loudly, boastfully; proudly, haughtily, pompously", we're given
a1500 Sidrak & Bokkus (Laud) (1999) II. 9225 Tonge begynneþ to waxe biglye And of his youthe he bosteth an hye.
1532 T. More Confut. Tyndale in Wks. 397/1 And bereth it out bigly wt shameles deuelyshe heresie.
1585 Abp. E. Sandys Serm. v. 89 Goliah thought bigly of himselfe.
1596 W. Warner Albions Eng. (rev. ed.) ix. xlvi. 218 Oftentimes Authoritie lookes biglier than a Bull.
1669 G. Burnet Modest Conf. between Conformist & Non-conformist (ed. 2) iii. sig. D1v, You talked bigly of jus divinum yet you minded it as little as any could.
1693 Dryden tr. Juvenal Satires 197 Bigly to look, and barb'rously to speak.
1744 Johnson Deb. in Gentleman's Mag. Feb. 69 Talking bigly indeed of vindicating foreign Rights.
1846 W. S. Landor Citation & Exam. Shakespere in Wks. II. 299 He spoke as bigly and fiercely as a soaken yeoman at an election feast.
1874 T. Hardy Far from Madding Crowd I. xxx. 331 ‘I don't see that I deserve to be put upon and stormed at for nothing!’ concluded the small woman, bigly.
So I'm sorry to conclude that Mr. Trump is not trying to bring back this estimable word. Rather, he's pushing an adverbial-adjunct extension of the phrase "big league".
This phrase started in the late 19th century as a way of referring to "the highest-ranking league in professional baseball, or, later, in other sports":
1882 Fort Wayne (Indiana) Daily Gaz. 8 Oct. 7/6 He will have a pitcher and catcher from one of the big league clubs.
1899 Sporting News 25 Mar. 1/3 This season marks an epoch in the matter of spring training for big League ball tossers.
And it soon acquired the obvious extended sense "of the highest rank; major, important, notable":
1917 N.Y. Tribune 21 Aug. 9/1 If it had put forward a big league candidate the interest in the campaign might have diverted public attention from the war.
1947 Time 14 Apr. 66/3 They announced a prize book contest baited with enough cash to make big-league authors sit up and take notice.
I'm not sure when "big league" was first used as a post-verbal adjunct, but it follows in the footsteps of "big time" in the sense "to a great degree, on a large scale, extremely", or which the OED has citations back to 1957:
1957 ‘E. Lacy’ Room to Swing vi. 93 The New York City police are good, big-time.
1983 Washington Post (Nexis) 27 Nov. c1 It's not like we're a powerhouse club that's rolling it up big-time.
1987 D. F. Wallace Broom of Syst. 416, I was big-time sad.
2006 Metro (Toronto) 11 Oct. 37/1 Wow, some of these people look like they just got beat down big time.
And this adverbial "big time" had its moment in the political spotlight (M.J. Stephey, "Bush's Major-League Mistake", Time 3/27/2012):
During the 2000 election campaign, microphones caught George W. Bush and Dick Cheney trashing Adam Clymer, a reporter for the New York Times. In a hushed whisper, Bush described Clymer as a "major-league asshole," to which Dick Cheney responded, "Yeah, big time." Later, Bush explained: "I said what I said. I'm a plainspoken fellow."
And adverbial "big league", in the same sense"to a great degree, on a large scale, extremely", has been around for a while. I'd be surprised if it didn't go back to the first half of the 20th century, though the earliest printed citations I've been able to find are much more recent, e.g.
[link 2004]: No way were Millennium's psychotherapies paying for even a fifth of it. Several investigative reporters had shown that. Outside interests had to be financing it big-league, especially given the Hard Times, and they'd require some kind of payoff.