The symbol $5!$ means $5\cdot 4\cdot 3\cdot 2\cdot 1$. What is the greatest odd integer that is a factor of $5!$ ?
Since the product of odd integers is another odd integer, the largest odd factor of 5!, or any number, is the product of all its odd prime factors. Clearly, the odd prime factors of 5! are 5 and 3, whose product is $5 \times 3 = \boxed{15}$.