Expresso started as a misspelling of espresso, which came to English from Italian and refers to a strong, pressure-brewed coffee. But because expresso has so often appeared in place of espresso, we can perhaps consider it a variant. And indeed, some dictionaries now list it as such. This doesn’t change the fact that many English speakers consider expresso wrong, however, and some will no doubt continue to do so no matter how common it becomes. So if you don’t want anyone to think you’re wrong, espresso is the safer choice.
It’s also worth noting that expresso is the French word for the pressure-brewed coffee, and this perhaps has had some small influence on English usage.
Although expresso remains far more common, examples such as these are not hard to find:
For their Christmas market, families can look forward to a Wiggles jumping castle, expresso coffee at $1, free express knife sharpening and a gold coin donation BBQ. [Southern Courier]
[S]itting down to order an expresso or a cappuccino was the height of cosmopolitan sophistication amongst the young at the time. [In Search of Fatima, Ghada Karmi]
This year, he imposed an “impolite” tax. The ubiquitous expresso, for which he charges 1.80 euros (R20), rises to two euros if customers forget to say “please”. [News 24]