What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?

$ 
\textbf{(A)}\ 760
\qquad\textbf{(B)}\ 756
\qquad\textbf{(C)}\ 755
\qquad\textbf{(D)}\ 752
\qquad\textbf{(E)}\ 750
$
1. **Understanding the problem**: We need to find the 111th smallest positive integer that does not contain the digits 3 or 4 in its base-5 representation. This means the digits of the number in base-5 can only be 0, 1, or 2.

2. **Counting in base-3**: Since we are restricted to the digits 0, 1, and 2 in base-5, we can think of this as counting in base-3. Each number in this sequence corresponds to a number in base-3.

3. **Finding the 111th number in base-3**: Convert 111 to base-3. 
   \[
   111_{10} \rightarrow \text{base-3}
   \]
   We perform the conversion by repeatedly dividing by 3 and recording the remainders:
   \[
   111 \div 3 = 37 \quad \text{remainder} \ 0
   \]
   \[
   37 \div 3 = 12 \quad \text{remainder} \ 1
   \]
   \[
   12 \div 3 = 4 \quad \text{remainder} \ 0
   \]
   \[
   4 \div 3 = 1 \quad \text{remainder} \ 1
   \]
   \[
   1 \div 3 = 0 \quad \text{remainder} \ 1
   \]
   Reading the remainders from bottom to top, we get:
   \[
   111_{10} = 11010_3
   \]

4. **Interpreting the base-3 number in base-5**: The base-3 number \(11010_3\) corresponds directly to a base-5 number since the digits 0, 1, and 2 are valid in both bases.
   \[
   11010_3 = 11010_5
   \]

5. **Converting \(11010_5\) to base-10**: Now, convert \(11010_5\) to base-10.
   \[
   11010_5 = 1 \cdot 5^4 + 1 \cdot 5^3 + 0 \cdot 5^2 + 1 \cdot 5^1 + 0 \cdot 5^0
   \]
   \[
   = 1 \cdot 625 + 1 \cdot 125 + 0 \cdot 25 + 1 \cdot 5 + 0 \cdot 1
   \]
   \[
   = 625 + 125 + 0 + 5 + 0
   \]
   \[
   = 755
   \]

Conclusion:
\[
\boxed{755}
\]