But two months after its debut, ChatGPT has more than 30 million users and gets roughly five million visits a day, two people with knowledge of the figures said. That makes it one of the fastest-growing software products in memory. (Instagram, by contrast, took nearly a year to get its first 10 million users.)

The growth has brought challenges. ChatGPT has had frequent outages as it runs out of processing power, and users have found ways around some of the bot’s safety features. The hype surrounding ChatGPT has also annoyed some rivals at bigger tech firms, who have pointed out that its underlying technology isn’t, strictly speaking, all that new.

ChatGPT is also, for now, a money pit. There are no ads, and the average conversation costs the company “single-digit cents” in processing power, according to a post on Twitter by Sam Altman, OpenAI’s chief executive, likely amounting to millions of dollars a week. To offset the costs, the company announced this week that it would begin selling a $20 monthly subscription, known as ChatGPT Plus.

Despite its limitations, ChatGPT’s success has vaulted OpenAI into the ranks of Silicon Valley power players. The company recently reached a $10 billion deal with Microsoft, which plans to incorporate the start-up’s technology into its Bing search engine and other products. Google declared a “code red” in response to ChatGPT, fast-tracking many of its own A.I. products in an attempt to catch up.

Mr. Altman has said his goal at OpenAI is to create what is known as “artificial general intelligence,” or A.G.I., an artificial intelligence that matches human intellect. He has been an outspoken champion of A.I., saying in a recent interview that its benefits for humankind could be “so unbelievably good that it’s hard for me to even imagine.” (He has also said that in a worst-case scenario, A.I. could kill us all.)

As ChatGPT has captured the world’s imagination, Mr. Altman has been put in the rare position of trying to downplay a hit product. He is worried that too much hype for ChatGPT could provoke a regulatory backlash or create inflated expectations for future releases, two people familiar with his views said. On Twitter, he has tried to tamp down excitement, calling ChatGPT “incredibly limited” and warning users that “it’s a mistake to be relying on it for anything important right now.”

He has also discouraged employees from boasting about ChatGPT’s success. In December, days after the company announced that more than a million people had signed up for the service, Greg Brockman, OpenAI’s president, tweeted that it had reached two million users. Mr. Altman asked him to delete the tweet, telling him that advertising such rapid growth was unwise, two people who saw the exchange said.