You are correct. You present a nuance of the 'working definition', a special case where "getting closer" is misleading. You should interpret "getting closer" as "getting as close as you like". This is a more accurate 'working definition' in any case. Then the constant function scenario works just fine. You can get as close as you like to the constant, quite trivially.
The "getting closer" definition makes it sounds as if a limit is somehow an indefinite thing, moving around, getting closer to things. This is misleading. A better intuition is "the limit at $x_0$ of $f$ is $L$" is "the value $f(x)$ can be made as close as you like to $L$ for all $x$ that is sufficiently close to but not equal to $x_0$".
First, congratulations on being an inquisitive mathematics student! These sorts of questions are one of the most important questions you should be asking yourself and others. Questions such as "why is this important?" and "why is this called that way?" are precisely what mathematics is about - i.e. it's not a set of arbitrary rules to annoy students, all the things got created for a reason!
Are there infinitely many boring limits?
You are correct, most "nice" functions defined on an interval or on $\mathbb R$ have infinitely many limits (see the comments section for counterexamples) and yes, they are "stupidly obvious" for continuous functions, i.e. $$\lim_{x\to a} f(x) = f(a)$$ which is often the definition of continuity, too.
But then there are many interesting cases, for example, try to figure out what's happening for $\lim_{x\to 0} \sin(1/x)$ and $\lim_{x\to 0} x\sin(1/x)$. Even if this is beyond your level, just thinking about the functions and looking at their graphs will give you some intuition about how interesting limits can be. One of them is continuous at zero. Which one? Why? What happens to the other one?
$$\sin(1/x)$$
$\sin(1/x)$" />
$$x\sin(1/x)$$
As a sidenote, there are also functions that are discontinuous everywhere. Those are usually rather hard to understand, though (althought Dirichlet's function is quite accessible)
Even "boring" limits are useful
But even for functions that are not that interesting, like $\text{sgn}(x)$, which gives you the sign $x$, i.e. it is $-1$ for negative numbers, $1$ for positive numbers and $0$ for zero, limits are a useful concept. The one-sided limits at zero (coming from left and right) are $-1$ and $1$, respectivelly, whereas the function value is $0$. This intuitively makes complete sense (draw it!) and thus having a rigorous mathematical object, the limit, to support this intuition is useful.
Why "limit"?
I do not know the proper etymology of the mathematical term "limit", but the english word comes from the word for "frontier", or "boundary". This makes sense even for a "boring" limit, like $\lim_{x\to 2} x^2$ - as you yourself suggested, you can approach it through the sequence $1.9,1.99,1.999,\dots$, that is you are coming closer and closer to the boundary point $2$, but never quite touch it, even though you're performing infinitely many steps. In that sense, $2$ would be your "limit", or the "boundary", which you never quite attain.
Lastly, note that you never actually touch the limit point while you're approaching the limit. This is important - what if the point was, say, undefined! Answers by other users stress this point more, make sure you read them.
Best Answer
If you do not like the word "approach" let us say the limit predicts where $f(x)$ will end up as $x \to c$.
If $f(x)$ sits there all along on its chair at $5$, surely it makes sense to predict it'll stay there.