Seemingly every week, Christophe Fouquet, the chief executive of the Dutch technology company ASML, has found himself grappling with political firestorms.

Last month, President Trump announced 50 percent tariffs on European goods sold to the United States, potentially raising some costs for ASML’s lithography machines, which are critical for producing advanced microchips. Two days later, he paused the tariffs.

Around the same time, the foreign minister of the Netherlands was in Beijing, partly to discuss lifting the rules that bar Chinese companies from buying ASML’s equipment. Then this week, the Dutch government collapsed, throwing any trade talks into question.

All of these events had the potential to disrupt ASML, which is the only maker of complex lithography machines that can cost as much as $400 million and be as big as a train car. The tools are so coveted by nations such as China and the United States for making cutting-edge chips that ASML has been turned into a geopolitical chess piece in trade battles, with some of its products restricted for export to certain countries.

The tussling has made it an uncertain time for Mr. Fouquet, 52, a Frenchman who took over as chief executive of the $300 billion company last year. “A large part of it is still out of our hands,” he said in a recent interview from ASML’s headquarters in the small town of Veldhoven, about 80 miles south of Amsterdam.