\text{The value of}\sqrt{77+\sqrt{12+\sqrt{10+\sqrt{32+\sqrt{16}}}}}\text{ is:}

The correct option is 9.

To find the value of the given nested radical expression, we simplify it step-by-step from the innermost square root to the outermost square root.

The original expression is:

$$\sqrt{77+\sqrt{12+\sqrt{10+\sqrt{32+\sqrt{16}}}}}$$

Step 1: Simplify the innermost square root
The innermost square root is:
$\sqrt{16}=4$
Substitute this back into the expression:
$$\sqrt{77+\sqrt{12+\sqrt{10+\sqrt{32+4}}}}$$

Step 2: Simplify the next innermost square root
Add the terms inside the radical:
$32+4=36$
Now evaluate the square root of 36:
$\sqrt{36}=6$
Substitute this value back into the expression:
$$\sqrt{77+\sqrt{12+\sqrt{10+6}}}$$

Step 3: Simplify the next level
Add the terms:
$10+6=16$
Now evaluate the square root of 16:
$\sqrt{16}=4$
Substitute this back:
$$\sqrt{77+\sqrt{12+4}}$$

Step 4: Simplify the next level
Add the terms:
$12+4=16$
Evaluate the square root:
$\sqrt{16}=4$
Substitute this back:
$$\sqrt{77+4}$$

Step 5: Final calculation
Add the remaining numbers:
$77+4=81$
Find the final square root:
$\sqrt{81}=9$

Therefore, the value of the given expression is 9.