Katherine makes Benj play a game called $50$ Cent. Benj starts with $\$0.50$, and every century thereafter has a $50\%$ chance of doubling his money and a $50\%$ chance of having his money reset to $\$0.50$. What is the expected value of the amount of money Benj will have, in dollars, after $50$ centuries?
1. Let \( e_n \) be the expected amount of money that Benj will have after \( n \) centuries where \( n \in \mathbb{Z}_{\geq0} \). We wish to find \( e_{50} \).

2. From the given information, we have the following recurrence relation for the expected value:
   \[
   e_n = \frac{1}{2}(2e_{n-1}) + \frac{1}{2} \cdot 0.50
   \]
   Simplifying this, we get:
   \[
   e_n = e_{n-1} + 0.25
   \]

3. To solve this recurrence relation, we can use induction. We start with the base case:
   \[
   e_0 = 0.50
   \]

4. Assume that \( e_n = e_0 + n \cdot 0.25 \) holds for some \( n \). Then for \( n+1 \), we have:
   \[
   e_{n+1} = e_n + 0.25
   \]
   By the induction hypothesis:
   \[
   e_{n+1} = (e_0 + n \cdot 0.25) + 0.25 = e_0 + (n+1) \cdot 0.25
   \]
   Thus, the induction step holds, and by induction, we have:
   \[
   e_n = e_0 + n \cdot 0.25 \quad \forall \ n \in \mathbb{Z}_{\geq0}
   \]

5. Substituting \( n = 50 \) into the formula, we get:
   \[
   e_{50} = e_0 + 50 \cdot 0.25
   \]
   Given \( e_0 = 0.50 \):
   \[
   e_{50} = 0.50 + 50 \cdot 0.25 = 0.50 + 12.50 = 13.00
   \]

The final answer is \(\boxed{13}\) dollars.