For intuition, what are some real life examples of uncorrelated but dependent random variables? In explaining why uncorrelated does not imply independent, there are several examples that involve a bunch of random variables, but they all seem so abstract: 1 2 3 4.
This answer seems to make sense. My interpretation: A random variable and its square may be uncorrelated (since apparently lack of correlation is something like linear independence) but they are clearly dependent.
I guess an example would be that (standardised?) height and height$^2$ might be uncorrelated but dependent, but I don't see why anyone would want to compare height and height$^2$.
For the purpose of giving intuition to a beginner in elementary probability theory or similar purposes, what are some real-life examples of uncorrelated but dependent random variables?
 A: A simple example is a bivariate distribution that is uniform on a doughnut-shaped area. The variables are uncorrelated, but clearly dependent - for example, if you know one variable is near its mean, then the other must be distant from its mean. 
A: I found the following figure from wiki is very useful for intuition. In particular, the bottom row show examples of uncorrelated but dependent distributions.

Caption of the above plot in wiki:
Several sets of (x, y) points, with the Pearson correlation coefficient of x and y for each set. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.
A: There are two words that you mention in the title of your question that are usually used interchangeably, correlation and dependence, but in the body of your question you restrict the definition of correlation to Pearson correlation, which in my opinion is indeed the appropriate meaning to correlation, when no other detail is provided. However, I believe that what you really want to ask goes beyond linear correlation, towards statistical dependence, that is: When are variables dependent, but independent when measured?
I mean, it's straightforward that a measure of linear association won't catch an association between variables that are associated but not in a linear way. Examples of that are all around us, though a r value of exactly 0 can be hard to find.
However, going back to the broader question that I elaborated, there could be spurious independence. That is, the variables are dependent, but your sampling will suggest that they are independent. I wrote an article about this, and there are scientific papers mentioning this problem too, such as this one.
Controlling for variables can be equivalent to slicing your data. By slicing too much (adjusting for many other variables), it's expected for your two random variables to appear independent. One may say: But I am not adjusting for anything! And the answer is: You don't need to. The collected data may be biased (selection bias) and you're not aware of it.
A: In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns $r_t:=(P_t-P_{t-1})/P_{t-1}$, with $P_t$ the price at time $t$, themselves are uncorrelated with their own past $r_{t-1}$ if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares $r_t^2$ and $r_{t-1}^2$ are not: there is time dependence in the variances, which cluster in time, with periods of high variance in volatile times.
Here is an artificial example (yet again, I know, but "real" stock return series may well look similar):

You see the high volatility cluster around in particular $t\approx400$.
Generated using R code:
library(TSA)
garch01.sim <- garch.sim(alpha=c(.01,.55),beta=0.4,n=500)
plot(garch01.sim, type='l', ylab=expression(r[t]),xlab='t')

