largest fraction less than 1 What is the mathematically rigorous way to answer the question: "what is the largest fraction less than 1"? (or to explain why it cannot be answered in the manner worded).

There is no such fraction. 

To see that this is the correct answer, note that for any fraction $\frac{p}{q}$ less than one, there is a slightly bigger fraction which is still less than one; in particular,  $\frac{p+q}{2q}$ is a fraction such that $\frac{p}{q} < \frac{p+q}{2q} < 1$. 
