Why does independence imply zero correlation? First of all, I'm not asking this:
Why does zero correlation not imply independence?
This is addressed (rather nicely) here: https://math.stackexchange.com/questions/444408/why-does-zero-correlation-not-imply-independence
What I'm asking is the opposite...say two variables are entirely independent of one another. 
Couldn't they have a tiny bit of correlation by accident?
Shouldn't it be...independence implies VERY LITTLE correlation?
 A: By the definition of the correlation coefficient, if two variables are independent their correlation is zero. So, it couldn't happen to have any correlation by accident!
$$\rho_{X,Y}=\frac{\operatorname{E}[XY]-\operatorname{E}[X]\operatorname{E}[Y]}{\sqrt{\operatorname{E}[X^2]-[\operatorname{E}[X]]^2}~\sqrt{\operatorname{E}[Y^2]- [\operatorname{E}[Y]]^2}}$$
If $X$ and $Y$ are independent, means $\operatorname{E}[XY]= \operatorname{E}[X]\operatorname{E}[Y]$. Hence, the numerator of $\rho_{X,Y}$ is zero in this case.
So, if you don't change the meaning of correlation, as mentioned here, it is not possible. Unless,  clarify your defintion from what the correlation is.
A: Comment on sample correlation. In comparing two small independent samples
of the same size, the sample
correlation is often noticeably different from $r = 0.$ [Nothing here contradicts @OmG's Answer (+1) on the population correlation $\rho.]$
Consider correlations between a million pairs of independent
samples of size $n = 5$ from the exponential distribution with rate $1.$
set.seed(616)
r = replicate( 10^6, cor(rexp(5), rexp(5))  )
mean(abs(r) > .5)
[1] 0.386212
mean(r)
[1] -0.0005904455

hist(r, prob=T, br=40, col="skyblue2")
  abline(v=c(-.5,.5), col="red", lwd=2)


For example, here is the scatterplot of first of the million pairs of samples of size $5,$ for which
$r =  -0.5716.$

There is nothing special about the exponential distribution in this regard.
Changing the parent distribution to standard uniform gave the following results.
set.seed(2019)
...
mean(abs(r) > .5)
[1] 0.391061
mean(r)
[1] 1.43269e-05


By contrast, here is the corresponding histogram of correlations for
pairs of normal samples of size $n = 20.$

Note: Other pages on this site discuss the distribution of $r$ in more detail; one of them is this Q & A.
A: Simple answer: if 2 variables are independent, then the population correlation is zero, whereas the sample correlation will typically be small, but non-zero.
That is because the sample is not a perfect representation of the population.
The larger the sample, the better it represents the population, so the smaller the correlation you'll have. For an infinite sample, the correlation would be zero.
A: Maybe this is helpful for some people sharing the same intuitive understanding. We've all seen something like this:

These data are presumably independent but clearly exhibit correlation ($r = 0.66$). "I thought independence implies zero correlation!" the student says.
As others have already pointed out, the sample values are correlated, but that does not mean the population has nonzero correlation.
Of course, these two should be independent—given Nicolas Cage appeared in a record-setting 10 films this year, we shouldn't be closing the local pool for the summer for safety purposes. 
But when we check how many people drown this year, there is a small chance that a record-setting 1000 people drown this year.
Getting such correlation is unlikely. Maybe one in a thousand. But it's possible, even though the two are independent. But this is just one case. Consider that there the millions of possible events to measure out there, and you can see the chance that the odds of some two happening to give a high correlation is quite high (hence the existence of graphs such as that above). 
Another way to look at it is that guaranteeing that two independent events will always give uncorrelated values is itself restrictive. Given two independent dice, and the results of the first, there are a certain (sizable) set of results for the second dice which will give some nonzero correlation. To restrict the second dice's results to give zero correlation with the first is a clear violation of independence, as the first dice's rolls are now affecting the distribution of the results.
