A few years ago I visited 16 countries in 6 months - all of those in Europe but each one for at least five days. That's perfectly manageable (I was a student with lots of holiday but a small budget) and I think it probably counts as "seeing" a country.
Since then, however, my standards have changed and now I don't really count myself as having "seen" a country unless I've spent a good two weeks there. Obviously for some very small countries (Luxembourg for example) you might not want that long, but, even countries that we think of as "small", are actually quite big and have a lot to offer.
Let's take the example of Albania. I've met lots of people who have "been there" but usually only for a day-trip from Corfu or on an overnight bus with a day in Tirana. I've spent two and a half weeks there and seen a lot of the highlights but by no means all and I'd happily go back.
At the other extreme I've spent two weeks in the USA - NYC, Las Vegas, Orlando. But there's no way I count myself as having "seen" the whole of the US!!
If you just want to set foot in a country then I reckon a year and a lot of money and a lot of patience at certain embassies and borders and you could do every country.
But if you want to actually "enjoy" your holiday and really "see" each country and appreciate it then I recommend setting this as a life-long goal.
Enjoy the journey - I know I will.
