One day in late November, a surprise caller turned up at Perrigo Co Plc’s Dublin headquarters, close to the city’s so-called Googletown neighborhood.

Informed that the drug maker’s Chief Financial Officer Ron Winowiecki wasn’t around, the visitor dropped off a letter at its second-floor office in the Treasury Building, famed for “Aspiration,” the sculpture of a naked woman scaling its exterior walls. Inside the envelope was a tax demand for a whopping 1.6 billion euros ($1.8 billion), the second-largest in Irish history, according to an account of the episode given to the government.

Weeks later, Perrigo revealed the assessment to...