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When Mr. Altman visited TSMC’s headquarters in Taiwan shortly after he started his fund-raising effort, he told its executives that it would take $7 trillion and many years to build 36 semiconductor plants and additional data centers to fulfill his vision, two people briefed on the conversation said. It was his first visit to one of the multibillion-dollar plants.

TSMC’s executives found the idea so absurd that they took to calling Mr. Altman a “podcasting bro,” one of these people said. Adding just a few more chip-making plants, much less 36, was incredibly risky because of the money involved.