What is a simple example of a limit in the real world?

Your example of a limit is of a limit which is easy to evaluate, but it's still a perfectly reasonable example!

Here's another fairly easy to grasp example of a limit which avoids triviality.

If I keep tossing a coin as long as it takes, how likely am I to never toss a head?

Rephrased as a limit problem, we might say

If I toss a coin $N$ times, what is the probability $p(N)$ that I have not yet tossed a head? Now what is the limit as $N\to\infty$ of $p(N)$?

The mathematical answer to this is $p(N)=\left(\frac{1}{2}\right)^N$. Then $$\lim_{N\to\infty}p(N) = 0$$ because $p=\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$ gets closer and closer to zero as $N$ gets "closer to $\infty$".

The reading of your speedometer (e.g., 85 km/h) is a limit in the real world. Maybe you think speed is speed, why not 85 km/h. But in fact your speed is changing continuously during time, and the only "solid", i.e., "limitless" data you have is that it took you exactly 2 hours to drive the 150 km from A to B. The figure your speedometer gives you is at each instant $t_0$ of your travel the limit $$v(t_0):=\lim_{\Delta t\to0}{s(t_0)-s(t_0-\Delta t)\over\Delta t}\ ,$$ where $s(t)$ denotes the distance travelled up to time $t$.

It is hard for me to stray from the confines of mathematics to the 'real world', so let me give you this "example":

Limits are super-important in that they serve as the basis for the definitions of the 'derivative' and 'integral', the two fundamental structures in Calculus! In that context, limits help us understand what it means to "get arbitrarily close to a point", or "go to infinity". Those ideas are not trivial, and it is hard to place them in a rigorous context without the notion of the limit. So more generally, the limit helps us move from the study of discrete quantity to continuous quantity, and that is of prime importance in Calculus, and applications of Calculus.

To apply this notion to physics (yes, I'm moving away from math now), it is possible to apply a continuous analysis to motion. We'd like to be able to measure instantaneous speed, which requires the notion of an instantaneous value. Now this is dependent on the concept of the limit. That is to say, we want to measure a quantity in an instant, and we define this "instant" by a limit, i.e., as an approach towards some infinitesimal time. This is how we would answer, e.g., the commonplace question "how fast was he going at time $x$?".