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It's been a very long time since I've done musical theory but I know this rhythm is quite common, especially in dance and even some punk/metal.
So, some songs have 4 beats to a bar but it has 2 triplets and 1 double.
In total, it consists of 8 notes however there are 3 accented notes. Like this:
1 2 3, 1 2 3, 1 2
It's not 4/4 (although you can count 4 beats) and it's not 6/8. (2 triplets to a bar).
Is it 8/8? What's the time signature?
1 2 3, 1 2 3, 1 2 is Calypso rhythm.
Although it appears it often has a 4/4 or 8/8 time signature, I have seen it 3+3+2/8. Even with a more regular time signature, you may find it notated with two dotted-crotchets (which shouldn't cross over the secondary beat on to the third crotchet) and a dotted bar-line before the fourth crotchet.
Yes, that would be 8/8. Mathematically it is the same with 4/4, but it differs on the accented beats. Where 4/4 would be:
1 2 3 4
8/8 is:
1 2 3, 1 2 3, 1 2
like the one in the song you provided.
It's a very common pattern, and it can (and probably should) definitely be notated in 4/4. It is the first half (the "three-side") of the traditional clave pattern.
It's called a tresillo. As others have mentioned, it's the first half of many other rhythms, but the tresillo is an important rhythm in its own right.
The original question specifies time signature.
Shouldn't the answer show an additive time signature?
The rhythm is common and can be expressed in notation with beaming or accent marks, etc. But there also is a time signature to represent it metrically.
Here is an example from Bartok's Mikrokosmos...
Well, I cannot view the video in Germany but { 4. 4. 4 } (two dotted crotchets and one normal crotchet) is a common syncopated rendition of the "original" { 2 4 4 } (one minim and two crotchet) rhythm of 4/4 for tango. In fact, if you are not playing old arrangements of old tangos, you are much more likely to get the syncopated version these days, particularly in Tango Argentino.
The pattern of 3,3,2 is the first bar of the two bar sequence Bossa Nova. Generally written in 4/4. The 'ones' are accented, often on a snare drum, with the stick on the skin, tapping the rim. There's probably a proper name for that... if not, there's another question coming...
When there are 4 beats in a bar and you have 8 quavers, the rhythms that you're talking about, despite being in groups of 3, aren't really triplets. Triplets are usually 3 quavers squeezed evenly into a space that would usually only house 2. If they were triplets, you would only have 3 beats in the bar and would be in 3/4 time - a different rhythm and feel to what I think you're describing.
In your song you've got a syncopated rhythm, 8 quavers with emphasis on the beats in bold:
1 (+2) + (3+) 4 (+)
When music is written down, the purpose of the time signature is to make it clear to the musician reading the piece (who may never have heard it) where the accents fall and how the groove is generally grouped.
With that in mind, you could write it in 4/4 time and most people would say that it would be correct, but I agree with the previous poster who said 3+3+2/8 It makes it much clearer what the feel of the song is - but then you could write it in 4/4 and make it clear in the way you group the notes; as long as it's clear where the emphasis is it doesn't really matter.
Things like that matter less when you're playing together and everyone knows the feel, and more when you're trying to record or play to a metronome. For example, when you're playing to a click track and the clicks are just going 1, 2, 3, 4 - that 4/4 feel is quite off-putting. I just had this experience recording a guitar arrangement of Stand By Me where the bassline is syncopated. I ended up recording a loop of the 123 123 12 rhythm to use as a click track to keep the feel.
Considering the accents as you write the sequence, you could also consider it like a couple of triplets in a | 2/4 | + | 2/8 | (no triplets in this last one... simply have to respect the same tempo's metronome). But being it not a regular rhythm, consider @MichaelCurtis' example the best answer. Anyway, the way you consider accents make the difference in the way you will execute/write it. There is not a real limit in it, but only the sound result you want obtain.
In addition to the other answers given, Bluegrass music typically follows a 3-3-2 pattern as well, usually picked out on the banjo.
