What is the largest number you can write with three $3$’s and three $8$’s, using only symbols $+,-,/,\times$ and exponentiation?
To determine the largest number that can be written using three 3's and three 8's with the operations \( +, -, \times, / \), and exponentiation, we need to explore the potential combinations and their results.

1. **Consider the use of exponentiation**:
   Exponentiation grows much faster than multiplication, addition, or subtraction. Therefore, we should prioritize using exponentiation to achieve the largest possible number.

2. **Formulate potential expressions**:
   We can form expressions like \( 3^{3^{3^{8^{8^{8}}}}} \) or \( 8^{3^{3^{3^{8^8}}}} \). However, we need to ensure that we are using exactly three 3's and three 8's.

3. **Evaluate the expressions**:
   Let's evaluate the potential expressions step-by-step to see which one yields the largest number.

   - **Expression 1**: \( 3^{3^{3^{8^{8^{8}}}}} \)
     - This expression uses three 3's and three 8's.
     - The innermost exponentiation \( 8^{8^{8}} \) is already an extremely large number.
     - Raising 3 to the power of such a large number will result in an astronomically large number.

   - **Expression 2**: \( 8^{3^{3^{3^{8^8}}}} \)
     - This expression also uses three 3's and three 8's.
     - The innermost exponentiation \( 8^8 \) is a large number, but not as large as \( 8^{8^{8}} \).
     - Raising 8 to the power of \( 3^{3^{3^{8^8}}} \) will also result in an extremely large number, but we need to compare it with the first expression.

4. **Compare the expressions**:
   - The expression \( 3^{3^{3^{8^{8^{8}}}}} \) involves a much larger exponentiation base compared to \( 8^{3^{3^{3^{8^8}}}} \).
   - Since \( 8^{8^{8}} \) is significantly larger than \( 8^8 \), the first expression \( 3^{3^{3^{8^{8^{8}}}}} \) will be larger.

Therefore, the largest number that can be written using three 3's and three 8's with the given operations is \( 3^{3^{3^{8^{8^{8}}}}} \).

The final answer is \( \boxed{3^{3^{3^{8^{8^{8}}}}}} \).