To determine which of the given numbers is irrational, we analyze each option step by step:

**Option A: $3.1415926$

This number is a finite decimal, meaning it can be expressed as a fraction of two integers. Thus, it is a rational number.

Option B: $-0.202002000$

Similar to option A, this number is also a finite decimal. Therefore, it can be expressed as a fraction, making it a rational number.

Option C: $\sqrt{25}$**

The square root of 25 is equal to 5, which is an integer. Integers are a subset of rational numbers since they can be expressed as a fraction (for example, $5 = \frac{5}{1}$). Thus, this option represents a rational number.

**Option D: $\sqrt[3]{9}$**

The cube root of 9 cannot be expressed as a fraction of two integers, making it an irrational number. Unlike the square root of a perfect square or integers, the cube root of a non-perfect cube does not simplify to a rational number.

Therefore, the correct answer, identifying the irrational number among the options, is:

$$\boxed{D}$$