This is one of the questions surrounding the word "couple", which derives from the Latin word copula (a bond or tie). (Yes, it is also the origin of the word "copulation"; please get your mind out of the gutter.) The word was first used in English (which borrowed it from French) in the 1300s, to designate a husband and wife, but it very quickly came to apply to two of anything.
You may be surprised to learn that some doughty "usage commentators" in the late 19th and early 20th century, undaunted by 5 centuries of usage, objected to "couple" being used to mean "two", saying that it originally meant the link between two things, not the things themselves. Not getting far with that objection and thus threatened with job loss, the commentators realized they had to think of other complaints, and latched onto the then new usage of "couple" to mean "a few" (although, usage commentators being what they are, some kept up the rearguard action against the "two" meaning" and thundered that "couple" should ONLY be used to mean "a few")! I personally use "couple" only to mean "exactly two" and am always a little perplexed by the "few" usage, but that doesn't mean that I think everyone should be like me. No doubt other people are equally perplexed when they tell me they'd like a couple of jellybeans and I dole out a measly two to them!
But is it "a couple of jellybeans" or "a couple jellybeans"?
In North America, "couple" underwent a development parallel to what happened with other words designating "more than one": "hundred" and "dozen". Way back in Anglo-Saxon times, we couldn't say, "He had a hundred sheep", because "hundred" is a noun. So the Anglo-Saxons had to say "He had a hundred of sheep" (or they had the option, which English no longer has, of using the genitive form of the noun "sheep", which amounts to the same thing). This carried on well into the 1600s. "Hundred"'s origin as a noun is still evident in the fact that we have to say "a hundred things" or "one hundred things" rather than just "hundred things" as we would with other cardinal numbers.
The same thing happened with "dozen" (which we borrowed from Old French dozeine, from the Latin for "twelve", duodecim). As in French today, where, if you want a dozen eggs, you have to say "une douzaine d'oeufs", not "une douzaine oeufs", originally we said "a dozen of eggs" (and indeed we still have to say "dozens of eggs"). This carried on well into the 1700s, but by the mid-19th century, the adjectival use of "dozen" took over from the noun usage, as this Google Books ngram comparing "a dozen eggs" and "a dozen of eggs" shows.
Not surprisingly, then, "couple" started along the same path, but a little later, apparently around the late 19th century. Being more recent, and, to make matters worse in the eyes of some people, originally North American to boot, this development is still raising hackles. "A couple things" is already very common in speech and quite common in informal prose. In view of this frequency (and the history of "hundred" and "dozen"), this usage will probably eventually take over. There are really no grounds for complaining about it, other than that it is relatively new, and different from what one might oneself say. But those are not objective grounds for complaint. For now, you probably want to include the "of" if you are writing formal prose, but bear in mind that a hundred years from now someone may look at your deathless prose and find it as quaint as we do now looking at 19th-century recipes that call for "a dozen of eggs".
PS: See an update to this post here.
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