How is it possible to define differential privacy on two databases that differ more than a single entry? The original definition of $\epsilon-$differential privacy is for two databases $D_1$, $D_2$ that differ at most one entry and an randomized algorithm $A$.

We have a bound on the probability ratio
$\frac{Pr[A(D_1) \in S]}{Pr[A(D_2) \in S]} \leq e^\epsilon$ where $S$ is subset of range of $A$.

How do we define such thing for two databases that differ more than one entry?
One of the advantages of differential privacy is composition. That is, if $D_1$ and $D_k$ differ on $k$ entries, then $k\cdot\epsilon$ differential privacy is achieved. This is easily shown by writing a series of equations. Specifically, let $D_1$ and $D_k$ differ on $k$ entries, and let $D_i$ be a database that differs from $D_{i-1}$ on one entry (for $i=2,...,k$). Then,
$$
\frac{\Pr[A(D_1)\in S]}{\Pr[A(D_k)\in S]} = \frac{\Pr[A(D_1)\in S]}{\Pr[A(D_2)\in S]} \cdot \frac{\Pr[A(D_2)\in S]}{\Pr[A(D_3)\in S]}\cdots \frac{\Pr[A(D_{k-1})\in S]}{\Pr[A(D_k)\in S]} \leq (e^\epsilon)^k = e^{k\cdot\epsilon}
$$
where each
$$
\frac{\Pr[A(D_{i-1})\in S]}{\Pr[A(D_i)\in S]} \leq e^\epsilon
$$
by the assumption of differential privacy for databases that differ on a single entry.