Lebniz's formula is an infinite sum of fractions, meaning we are dealing with rational numbers everywhere in it. If you start to approximate π (π over 4) by calculating the partial sums of the formula, you can keep using fractions - it's all just the matter of finding the least common multiple of the denominators and thus expanding the fractions.
The approximation goes like 1, 2/3, 13/15, 76/105, ...
Theoretically, you can repeat this fraction expansion infinite number of times. And no matter how long you're doing it, you will always get a rational number.
But if you never stop, you do it all the way down to infinity, the result is π over 4, which is insanely irrational. What is the point where (and the reason why) this rational pattern breaks?
There is no such point. This is an instance of the fact that any real number (rational or not) can be expressed as the limit of a sequence of rational numbers.
A similar situation occurs with the sequence $\left(\frac1n\right)_{n\in\mathbb N}$. Each term of the seuence is greater than $0$. But the limit of the sequence is $0$. When do the terms of the sequence become $0$? Never!