When $(10^5 \cdot 10^{77} \div 10^{15} )\div \left(10^{15}\right)^4$ is written as a single integer with no exponents, it is a 1 followed by one or more zeroes. How many zeroes are there?
If we can express this number as a single power of 10, then we can directly find the number of zeroes at the end of the numbers. Recall that $10^n$ has $n$ zeroes at the end of it. We must start with parentheses first. Recall that $\left( a^m \right) ^n = a^{mn}$. Because of this, we can rewrite the last term as $10^{15 \cdot 4}=10^{60}$. Finally, recall the rules of division and multiplication of exponents: $a^m \cdot a^n = a^{m+n}$ and $a^m \div a^n = a^{m-n}$. Using these we can carry out the multiplication and division from left to right to get \begin{align*}
(10^5 \cdot 10^{77} \div 10^{15}) \div \left(10^{15}\right)^4 &= (10^5 \cdot 10^{77} \div 10^{15}) \div 10^{60} \\
&=(10^{5+77} \div 10^{15}) \div 10^{60} \\
&=(10^{82} \div 10^{15} )\div 10^{60} \\
&=10^{82-15} \div 10^{60} \\
&=10^{67} \div 10^{60} \\
&=10^{67-60} \\
&=10^7.
\end{align*}Because the exponent of the number 10 is 7, this number has $\boxed{7}$ zeroes at the end.