A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \geq 1, a_{n+1}$ is the decimal period of $1/a_n$ multiplies by $2$. Determine $a_{2003}.$
1. **Define the sequence and initial conditions:**
   - The sequence \((a_n)\) is defined such that \(a_1 = p\), where \(p\) is a prime number with exactly 300 nonzero digits.
   - For each \(n \geq 1\), \(a_{n+1}\) is the decimal period of \(1/a_n\) multiplied by 2.

2. **Understanding the period of \(1/p\):**
   - The period of \(1/p\) is the repeating part of its decimal expansion.
   - For example, if \(p = 7\), then \(1/7 = 0.\overline{142857}\), and the period is 142857.

3. **Determine the period length:**
   - Let \(n\) be the order of 10 modulo \(p\). This means \(n\) is the smallest positive integer such that \(10^n \equiv 1 \pmod{p}\).
   - The decimal expansion of \(1/p\) will repeat every \(n\) digits.

4. **Express \(1/p\) in terms of its period:**
   - We can write \(1/p\) as \(\frac{10^n - 1}{p \cdot 10^n - 1}\), where the numerator represents the repeating part of the decimal expansion.

5. **Calculate \(a_2\):**
   - The period of \(1/p\) is \(\frac{10^n - 1}{p}\).
   - Therefore, \(a_2 = 2 \cdot \frac{10^n - 1}{p}\).

6. **Determine \(a_3\):**
   - \(a_3\) is double the period of \(1/a_2\).
   - Since \(a_2 = 2 \cdot \frac{10^n - 1}{p}\), we need to find the period of \(1/a_2\).

7. **Simplify the expression for \(a_3\):**
   - Notice that \(a_2 = 2 \cdot \frac{10^n - 1}{p}\) can be written as \(\frac{2(10^n - 1)}{p}\).
   - The period of \(1/a_2\) is the same as the period of \(1/(2(10^n - 1)/p)\), which simplifies to \(p/(2(10^n - 1))\).

8. **Verify the periodicity:**
   - Since \(p\) is a prime number with 300 nonzero digits, it is much larger than \(10^n - 1\).
   - Therefore, the period of \(1/a_2\) is essentially the same as the period of \(1/p\), but scaled by a factor of 2.

9. **Establish the periodicity of the sequence:**
   - From the above steps, we see that \(a_3 = 10p\).
   - Since \(a_4 = a_2\), the sequence becomes periodic with a period of 2 starting from \(a_2\).

10. **Determine \(a_{2003}\):**
    - Since the sequence is periodic with a period of 2, \(a_{2003}\) will be the same as \(a_3\).

Conclusion:
\[
a_{2003} = 10p
\]

The final answer is \(\boxed{10p}\)