\text{The value of}\sqrt{17+\sqrt{57+\sqrt{45+\sqrt{7+\sqrt{81}}}}}\text{ is:}

The correct option is 5.

To find the value of the given nested radical expression, we simplify it step-by-step, starting from the innermost square root and working our way outwards.

The original expression is:

$$\sqrt{17+\sqrt{57+\sqrt{45+\sqrt{7+\sqrt{81}}}}}$$

Step 1: Simplify the innermost square root
We begin with the innermost term, which is $\sqrt{81}$. Since 81 is a perfect square ($9^2=81$), we get:
$$\sqrt{81}=9$$

Now, substitute this value back into the expression:
$$\sqrt{17+\sqrt{57+\sqrt{45+\sqrt{7+9}}}}$$

Step 2: Simplify the next innermost term
First, sum the numbers inside the radical: 7 + 9 = 16. The expression becomes:
$$\sqrt{17+\sqrt{57+\sqrt{45+\sqrt{16}}}}$$

Since 16 is a perfect square ($4^2=16$), we simplify $\sqrt{16}$ to 4:
$$\sqrt{17+\sqrt{57+\sqrt{45+4}}}$$

Step 3: Continue working outwards
Sum the terms inside the next radical: 45 + 4 = 49. The expression becomes:
$$\sqrt{17+\sqrt{57+\sqrt{49}}}$$

Since 49 is a perfect square ($7^2=49$), we simplify $\sqrt{49}$ to 7:
$$\sqrt{17+\sqrt{57+7}}$$

Step 4: Simplify the remaining internal nested radical
Sum the terms: 57 + 7 = 64. The expression becomes:
$$\sqrt{17+\sqrt{64}}$$

Since 64 is a perfect square ($8^2=64$), we simplify $\sqrt{64}$ to 8:
$$\sqrt{17+8}$$

Step 5: Final calculation
Now, add the final two values: 17 + 8 = 25. The expression is now simplified to:
$$\sqrt{25}$$

Since 25 is a perfect square ($5^2=25$), the final result is:
$$5$$