However, if you normalize your data to [0,1], you'll never have to compute 10^100 and thus never face any numerical issues. "Never" assumes no distribution shift.

Indeed, the examples work thanks to this choice of the interval, but this comes with the choice of the basis. Of course Bernstein basis functions explode outside [0,1], but I think the point is that high-degree polynomials pose no danger if you scale the data *according to the polynomial* (use [0,1] for Bernstein and [-1,1] for Chebyshev, for example). So the "magic combo" is polynomial + scaling to its interval. Otherwise all bets are off, of course.