Convergence with the $5$-adic metric. I'm struggling to get any intuition for the following example:
Consider the sequence $\{a_k\} = \{3, \ 33, \ 333, \ 3333, \ \ldots\}$
It's easy to show, using the geometric series formula, that $a_k = \frac{1}{3}(10^k-1)$.
It follows that $3a_k = 10^k-1$ and hence $3a_k+1=10^k$, and so $3a_k+1 \equiv 0 \bmod 5^k$.
By the definition of the $5$-adic metric, we have $0 \le |3a_k+1|_5 \le 5^{-k}$.
As $k \to \infty$, $5^{-k} \to 0$ and so $|3a_k+1|_5 \to 0$. We conclude that $a_k \to -\frac{1}{3}$ ($5$-adically).
How can a sequence of positive integers tend to a negative fraction?
It seems that $|a_k|_5 = 1$ for all $k$, but then $a_k \to -\frac{1}{3}$?
 A: Promoting some of the comments to an answer with a view of A) removing this from the list of unanswered questions, B) covering some of the features of the $p$-adic metric that surprise learners at first, and C) also aiming to draw an analogy with the ring of formal power series.
1. The $p$-adic "size" of rational numbers does not match the intuition based on the archimedean valuation better known as the absolute value. $2$-adically a high power of $2$ is tiny in comparison to a meager $-1$. Therefore, for example, the sequences $2^n\to0$ and $2^n-1\to-1$. In other words, in a series like
$$1+2+4+8+16+32+\cdots$$ the first term $1$ is the dominant one. This is not atypical of converging series in all domains :-)
2. The formula for the sum of a geometric series is your friend. Whenever it converges the sum of the series is surely (by the usual argument)
$$
a+aq+aq^2+aq^3+\cdots=\frac a{1-q}.\qquad(*)
$$
Here it converges if and only if the ratio $q$ has $p$-adic size $<1$ (hardly a surprise!). In other words, the formula $(*)$ holds iff $|q|_p<1$. Therefore, as Fly by Night observed
$$
...33333=3+30+300+3000+30000+\cdots=\frac3{1-10}
$$
whenever $|10|_p<1$. This inequality holds when $p=5$ or $p=2$, and in both cases the sum of this series is $-1/3$.
3. The mildly surprising fact about this limit is that a sequence of positive rationals has a negative limit. This will cease to amaze a learner whenever they recall that $2$-adically $1024$ is very close to $-1024$, but relatively far away from $1025$. A more formal way of phrasing this is that the $p$-adic fields do not have a total ordering - a relation that would allow us to, among other things, partition the $p$-adic numbers into negative and positive numbers. One rigorous argument for that parallels the reasoning why we don't have a total ordering in $\Bbb{C}$ either. Remember that if $i$ were either positive or negative, then its square should be positive, which it ain't. Similarly in all $p$-adic fields some negative integers have square roots. There is a $5$-adic $\sqrt{-1}$ (see here for a  crude description of the process of finding a sequence of integers converging $5$-adically to a number with square $=-1$. Exactly which integers have $p$-adic square roots is number-theoretic in nature. For example when $p=2$ it turns out that $\sqrt{m}$ of an odd integer $m$ exists inside $\Bbb{Q}_2$, iff $m\equiv1\pmod8$, so $-7$ has a $2$-adic square root.
4. The same themes recur, if we want to define a $p$-adic exponential function with the usual power series
$$
\exp(x):=\sum_{n=0}^\infty\frac{x^n}{n!}.
$$
The problem is with convergence. Contrary to expectations from real analysis this series will not converge for all $x\in\Bbb{Q}_p$. The reason is the denominators. For large $n$, the factorial will be divisible by higher and higher powers of $p$. Therefore we are dividing by a sequence of small numbers tending towards zero, and the numerator $x^n$ needs to compensate for that. A more careful analysis of the situation reveals, that this series converges, iff 
$|x|_p<p^{-1/(p-1)}$.
5. Apropos series. An analogue of the $p$-adic metric you may be familiar from courses on analysis is the $x$-adic topology (can turn it into a metric if so desired) on (formal) power series (with coefficients in, say, $\Bbb{R}$!). Two power series are close to each other $x$-adically, iff their difference is divisible by a high power of $x$. Therefore we can say that $e^x$ and $1+x+x^2/2$ are already quite close to each other, but $\sin x$ and $x-x^3/6+x^5/120$ are closer still. A common feature of all non-archimedean metrics ($p$-adic, $x$-adic,...) is that adding "small" numbers together will never make a large number, no matter how many of them you add together. So w.r.t. the $2$-adic metric no matter how many numbers divisible by four you add together you will never get a large number, say an odd number. Similarly, if you add together several formal power series divisible by $x$ you never create a non-zero constant term. This has an impact on some things. For example, when defining integrals, we want to approximate something by a sum of small things. In the $p$-adic world we need to...
