Where do harmonics come from? At first, I thought harmonics come from the signal being periodic. However, we know a sine wave is also periodic but contains a single frequency and no harmonics.
A harmonic for me is a component (pure sine-wave) with a frequency $f_k = k f_0$ for some $k\in \mathbb{Z}$ and a certain frequency $f_0$ called the fundamental frequency. To this, I add the condition that its amplitude is not zero.
We can also imagine the sum of two different frequencies, one is not necessarily the harmonic of the other, and get a periodic signal but no harmonics (according to my definition, which might need correction if it's wrong), just the two frequencies.
Below, the result of a simulation where I have the sum of two sine-waves with frequencies 100 Hz and 201 Hz, respectively:
$$
x(t) = \sin(2 \pi 100 t) + \sin(2 \pi 201 t)
$$
The signal is periodic (with a period equal to 1 s) but it does not contain harmonics (again according to my definition!).


Now, I'm wondering what's the real root-cause for harmonics appearing on a signal.
It appears that your understanding of harmonics is not entirely correct yet. Your example of two sinusoids with frequencies $f_1$ and $f_2$ can actually be explained in terms of harmonics IF
$$f_1=n_1f_0\quad\text{and}\quad f_2=n_2f_0,\qquad n_1,n_2\in\mathbb{Z^+}\tag{1}$$
is satisfied for some positive $f_0$. Note that this means that the ratio $f_1/f_2$ is rational. In that case the resulting sum is periodic. Choosing the smallest possible integer values of $n_1$ and $n_2$ such that $(1)$ is satisfied will allow you to compute the fundamental frequency $f_0$ of the resulting waveform.
In the case of your example we have $f_1=100$ and $f_2=201$ (units don't matter here). So we get $n_1=100$, $n_2=201$, and $f_0=1$ (same units as $f_1$ and $f_2$). So the sum is periodic with fundamental frequency $f_0=1$, even though the Fourier series coefficient of the fundamental is zero! Hence, we have a periodic signal with two harmonics and no fundamental. Nevertheless, the (fundamental) frequency of that signal equals $f_0=1$ (check that!).
With the period $T=1/f_0=1$ we have
$$x(t+T)=\sin\big[2\pi\cdot 100(t+T)\big]+\sin\big[2\pi\cdot 201(t+T)\big]=x(t)\tag{2}$$
where the last equality follows from the fact that we add an integer multiple of $2\pi$ to the argument of $\sin(x)$, which of course has period $2\pi$. Note that this doesn't show that $T$ is the smallest possible period, i.e., that $f_0$ is the largest possible fundamental frequency, but I trust that you can easily verify that.