Does this expression represent the largest real number? I'm not very good at this, so hopefully I'm not making a silly mistake here...
Assuming that $\infty$ is larger than any real number, we can then assume that:
$\dfrac{1}{\infty}$ is the smallest possible positive real number.
It then stands to reason that anything less than $\infty$ is a real number. Therefore, if we take the smallest possible quantity from $\infty$, we end up with:
$\infty-\dfrac{1}{\infty}$
Does that expression represent the largest real number? If not, what am I doing wrong?

Since $\infty$ is not a real number, you cannot assume that $\dfrac{1}{\infty}$ is a meaningful statement.  It is not a real number. 
You might want to investigate non-standard, hyperreal and surreal numbers and infinitesimals. 
