Weak buffers and accuracy when mixing acid and base
What is the weakest buffer possible - that is, what has the lowest buffer capacity possible?


You've conflated two different aspects of a buffer. First there is the $K_\mathrm{a}$, or $\mathrm{p}K_\mathrm{a}$ for the species.

$$K_\mathrm{a} = \frac{\ce{[H^+][A^-]}}{\ce{[HA]}}$$

$$\mathrm{p}K_\mathrm{a} = -\log{K_\mathrm{a}}$$

So weaker acids would have a high $\mathrm{p}K_\mathrm{a}$, close to 7 or greater. 

Then there is the buffer capacity which is a function of the absolute concentrations of the various species ($\ce{[H^+]}$, $\ce{[A^-]}$, and $\ce{[HA]}$). So if there is relatively much more $\ce{[H^+]}$ or $\ce{[A^-]}$ in the solution than acid or base being added, then the pH will change by only a small amounts. 


  Also, how precise do you have to be when mixing a strong acid and base to get a pH value of 7 and not 6?


If the strong acid is $10^{-6}$ greater than the strong base then the pH will be 6.

$\mathrm{pH} = -\log\ce{[H^+]}$