Concur with the 4-7 range given by others.
For me, there is significant overlap between “a few” and “several,” almost to the point of being synonymous, but “several” does feel like it extends farther than “a few.” “Seven-ish,” though, is on the high end of what I’m used to hearing “several” refer to. For me, it’s more like 5-ish. And “a couple (of)” is not literally only two. That tends to be the way I use it, but more often, at least how I encounter it, it refers to a small quantity greater than 1, so maybe 2-4 (though the literalists may insist it means two, despite the fact that in much of casual speech of average English speakers, it doesn’t). And I don’t really find it all that context dependent. I’m sure it is, but whether I’m talking on a scale of ten years or one thousand years, “several years” still means approximately the same thing to me. I can’t really think of an example where it would make that much a difference.
When I was a kid I thought “several” meant between “few” and “many,” but much closer to “many.” Then I learned it was less than I’d thought.
Now I use it to mean “a few.” The other day I referred to a package of three chocolate eggs as having “several” eggs.
Close. It’s certainly not close to “many,” but it’s more than a “few” but not quite as much as “a bunch.”
Except when it’s used in the slightly archaic sense of a set of separate things, as noted by others above, I assign “several” a value of 3 by default, but would not be surprised if it were as high as 5. Regardless of its precise value, it is an absolute term. “Few”, by contrast, is a term I regard as proportional. In general use, I treat it as almost interchangeable with “several” (i.e., in the absence of other information, I assign it a value of 3 or 4). However, when the context is things that come in larger quantities, “few” may refer to larger numbers, so long as they represent a small proportion of the total.
Say, for example, that I’ve been picking blackberries, and have accumulated 3 buckets full. I might say that I have “several buckets” or “a few buckets” of berries. When speaking of individual berries, however, “several” would still mean 3, while “a few” might mean a dozen.
2 is a couple
3 is a few
several is more flexible, generally in the 4-7 range.
What constitutes several?
2 Es, 1 A, & 4 consonants. 
I agree with the general thrust that context is important. Even something as seemingly exact as a “couple” can be vague. If I were to ask a friend if he had ever eaten at a particular restaurant and he casually replied “Yeah … couple times”, I wouldn’t necessarily assume that he had been there exactly twice. Or at least, if I later learned that he had been there four times in the last five years, I wouldn’t think he lied to me.
2=couple
3=few
4&5 = handfull
6-10 several
10-12= bunch
12=dozen
12+= many
24+=dozens
I wouldn’t balk on hearing someone use “several” for 12.
It’s not intended to be a precise term.
I have no problem referring to 12 as “several.” I can think of contexts where I wouldn’t use it, but I can think of a lot more contexts where 12 is too few to be considered “many.”
According to me, it means three or so. According to this Final Jeopardy, it means about 20.
Same here. ‘A few’ is intended to diminish the importance of what is being talked about, but the numbers are about the same as meant in ‘several’.
2 is a couple
3 is a few
4-7 is several
and after that it depends on what is being counted and that amount relative to what it could be.
I am one of the terrible sinners who uses “couple” this way. It drives my husband crazy.
‘A couple of’ can mean 2-3, maybe 4. ‘I’m going fishing with a couple of the guys’ does not exclude the possibility that 3 or 4 others are going, as the exact numer may not be known.
‘A pair’ is more precise.
if you are a kid then being allowed to have a few pieces of candy could amount to many.
To me, “several” and “a few” are synonyms meaning 3-7. Except I can never say “several” to a Thai, because everyone here thinks I’m saying “seven.”
The needs of the many outweigh the needs of the few or several.
Sadly this otherwise amusing link to pictures contains images of wild cats as hunting trophies, so people may prefer not to look.