Let's denote the number of people they can invite as 'x'.
The cost of the cleanup fee is $100, and the cost per person is $36. Therefore, the total cost is given by the equation:
$100 + $36*x = $3000
Subtracting $100 from both sides of the equation, we get:
$36*x = $2900
Dividing both sides of the equation by $36, we get:
x = $2900 / $36
Simplifying the right side of the equation, we get:
x = 80.56
Since the number of people must be a whole number, the greatest number of people they can invite is 80. Answer: \boxed{80}.