From a small office in Amsterdam, payment startup Adyen BV shows even under-the-radar companies can gain major traction in the booming world of payment processing.

The eight-year-old company quietly processed more than $14 billion in payment transactions last year from merchants including Groupon Inc., Evernote Inc. and Airbnb Inc. It reports earning $95 million in revenue last year, up from $65 million in 2012.

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