Break-up pi in whatever means you think is reasonable.
3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...)
Aka: 9.4245...
You know, how we've been multiplying 3 * pi for our whole lives. We split pi up into an infinite sum of component numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them together by individually multiplying the parts (3 * 3 + 3 * 0.1 + 3 * 0.04...)
I've responded to this elsewhere in the thread: I don't think adopting increasingly perverse interpretations of "repeated addition" as you try to include more and more numbers is a useful way to teach multiplication.
You extend to the rationals in the natural way, and then use continuity to define what happens for irrationals. (Of course, in a discrete context that doesn't apply. But the article's author is a grade school teacher.)
3 * pi = pi + pi + pi
but how do you represent the other distribution, where 3 is added together pi times?