Added later
I don't know why I am receiving so much upvotes, since the question is honest, have explained why is been done, and I put some examples of what kind of things I am thinking about could be candidates. For people that should like science, try to thinking "out of the box" should be something exciting, and not something insulting - by doing always the same things you will never push the borders of knowledge.
Recently I found on these papers here and here that there exists finite duration solutions to differential equations, and also that no Lipschitz ODE will support them, neither an analytic solution since at best they will vanish at infinity. So, if you do believe that there exist experiments which reaches and ending time, you must think of non-linearities of these kinds since no linear model neither non-piecewise power series will work.
Now, since everyday phenomena have ending times in my experience, let say a scalar system described by $r(t)$, since after the ending time $T$ it becomes zero forever, my intuition says that taking the variable $z(t)=1/r(t)$ it should behave as having a finite time blow up, and on the already linked question I found that it could be the case, but it also there is examples when it doesn't work (giving in the answers).
As a known example, the Norton's Dome which is kind of a rarity of system, its dimensionless equation $\ddot{r}=\sqrt{r}$ for real valued initial condition $r(0)=T^4/144>0$ and $r'(0)=-T^3/36<0$ it will stand finite duration solutions $r(t)=\frac{1}{144}(T-t)^4\theta(T-t)$ with $\theta(t)$ the standard Heaviside unitary step function (I think of this as pushing the ball from below the dome such it just stops at the higher point - but I am not fully sure if the right representation). Now, taking the system $z(t)=1/r(t)$ will lead to the differential equation $\ddot{z} +z\sqrt{z}-2\frac{(\dot{z})^2}{z}=0$ which indeed can stand a finite time blow up solution $z(t) = \frac{144}{(T-t)^4}$.
So after seeing a video where a worldwide expert is researching of blow ups in a wide known physical system as the Navier Stokes equation, and thinking that everyday phenomena indeed stops moving, I start wondering if there other kind of physics systems that behaves like having blow up that could be converted on finite duration solutions, or other finite duration models (in which I am interesting to find).
But as the example giving on this answer about the behavior of the collapsing bubble on somnoluminescence phenomena which radius behaves as $R(t) \sim (A-\alpha t)^{2/5}$ and $\dot{R}(t) \sim \frac{2}{5}\alpha (A-\alpha t)^{-3/5}$, and the Norton's Dome example also, they ends sharply as the square root function, I am looking if there are other examples where the singularity point is achieved "softly" (instead of "sharply"), as is done in the function of the question.
Since everyday phenomena haves ending times at first I thought it will be many examples, but looks like everything has been "linearized" in classic physics. Actually is disappointing to see all the negativity, but I still believe is not a loaded question but instead an interesting one.
And also I would like to thank to whom have already answer this question with really interesting and developed insights, I really appreciated.
