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What is the weakest buffer possible - that is, what has the lowest buffer capacity possible?

I would just assume that the lower the concentrations of $[\ce{A-}]$ and $[\ce{HA}]$, the lower the capacity is, but it seems like the question is looking for a specific solution?

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? since $[\ce{H+}] = 10^{-7}$ and $[\ce{OH-}] = 10^{-7}$ at $\mathrm{pH} = 7$, while $[\ce{H+}] = 10^{-6}$ and $[\ce{OH-}] = 10^{-8}$ at $\mathrm{pH} = 6$, does this imply you can at most be off by a factor of $10^{-2}$ mol/L?

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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.

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

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