I'd say that to call oneself rich, you'd need $5+ million in the bank.
Several years ago in a large U.S. newspaper, I read an article on the business page that explored the question, "How much money does it take to be a millionaire?"
The author explained that to most Americans, the word "millionaire" denoted extreme wealth. The word was coined back in the 1920s or '30s when a million bucks was a lot of money. Adjusted for inflation, he said, it would now take about $11 million in assets to live the "millionaire" lifestyle most people refer to when they use the word.
And that was at least seven or eight years ago, so I imagine it's $12 or $13 million today.