jmc

We begin by adding 2 to both sides of the equation, $$\begin{array}{rl}7& =x^2+\frac{1}{x^2}\\ \Rightarrow 9& =x^2+\frac{1}{x^2}+2\\ \Rightarrow 9& =x^2+2(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}\\ \Rightarrow 9& ={\left(x+\frac{1}{x}\right)}^2\end{array}$$ So, the possible values for $x+\frac{1}{x}$ are $3$ and $-3$. The greater of these is $\boxed{3}$.