jmc

If we are only interested in the units digit of a product of several numbers, we may drop any digits other than units digits as they will not affect the units digit of the product. Bringing in each factor one at a time, we find:

\begin{array}{r} The units digit of $\,7^1\,$ is 7, \\ $7\times7\,$ ends in 9, so the units digit of $\,7^2\,$ is 9, \\ $9\times7\,$ ends in 3, so the units digit of $\,7^3\,$ is 3, \\ $3\times7\,$ ends in 1, so the units digit of $\,7^4\,$ is 1, \\ $1\times7\,$ ends in 7, so the units digit of $\,7^5\,$ is 7, \\ $7\times7\,$ ends in 9, so the units digit of $\,7^6\,$ is 9, \\ $9\times7\,$ ends in 3, so the units digit of $\,7^7\,$ is $\,\boxed{3}$. \end{array}