Everyone pays airport taxes. Most people don't realize it, though, because these taxes are generally included in ticket prices. On a recent trip to Buenos Aires, though, I was reminded in a very direct fashion that some countries--Argentina among them, but also Costa Rica and dozens of other destinations across the developing world--prefer to collect their departure taxes themselves at the airport.
In most cases, the tax is $20-30 USD. What makes it fascinating is that it's almost always paid in dollars (or euros) rather than the local currency. This decision brings stable currency into government coffers but has the implicit impact of making leaving the country a variable expense for the nation's own citizens (who are paid in local currency and not dollars or euros). Handing over my last U.S. $20 bill to pay the tax was an annoyance; for an Argentine with tickets to leave the country, not having $60 AR pesos would've stopped the trip before it even began.
Bottom line: you can't leave without paying the departure tax. That means that you, as a Spontaneous Tourist, should be careful to always have $20-30 USD set aside should you need it at the last minute. Fortunately, in most cases, credit cards are accepted. Happy travels!