My favorite near miss not documented in the article comes from the golden ratio 1.618033988749895…
Repeated powers of the golden ratio are remarkably close to integers. By the time we raise the golden ratio to about the 72nd power, IEEE 754 double-precision floats cannot tell that it is not an integer.
On one hand, every power of the golden ratio is irrational. However, on the other hand, the golden ratio's powers encode the Fibonacci sequence as a recurrence relation, so that the powers tend to be integers. This tension is what creates the near misses.