Does there exist some other type of electromagnetic waves? When I learned about electromagnetic waves, I was told that some accelerating charge, specifically oscillating, produces electromagnetic waves, in a way like this:
It produces a changing magnetic field, which in turn produces a changing electric field, which in turn produces a changing magnetic field, which keeps on propagating in space, and called an electromagnetic wave.
Then I was told these fields are in form of sine and cosine waves, so it clicked like this, via the differentiation loop: 
$$
{+\sin} 
\longrightarrow {+\cos} 
\longrightarrow {-\sin} 
\longrightarrow {-\cos} 
\longrightarrow \cdots
.
$$
If charge oscillates as $\sin$, then magnetic field as $\cos$, electric field as $-\sin$, re-magnetic-field as $-\cos$, and so on.
But, we know about one more differentiation loop: $e^x \longrightarrow  e^x \longrightarrow  \cdots$. So if charge accelerates exponentially, will it produce an electromagnetic wave??
 A: Your qualitative understanding of EM waves is based on some facts, but it is too vague to allow you to make speculations.
EM waves are described by wave solutions of Maxwell's equations. Translation in words of the precise mathematical content of the equations and their solutions can be done, but with some care, to avoid to convey something which is not in the formulae. In particular, it could be source of misunderstanding to insist too much on a causal-relation of a magnetic field creating an electric one, creating a magnetic one ... and so on. Magnetic and electric field in an EM plane wave are exactly in phase and, at a fixed point of the space they vanish or get the maximum intensity at the same time.
Plane waves are very simple examples of waves, but not the most general ones. Moreover,  they are quite unphysical: strictly speaking they cannot exist, since a plane wave starts at $t=-\infty$ and lasts forever, with a signal (carrying energy) spread over infinite planes. Still, the importance of plane wave solutions is difficult to overestimate, due to the Fourier's theory which ensures that all the physical (bounded in time and space) waves can be formally described as superposition of (an infinite number of ) plane waves. These superpositions (wave-packets) do correspond to the measured waves.
Now, an elementary plane wave (i.e. the elementary constituent of a wave-packet)  can be described by a function of time $t$ and coordinates ($x,y,z$) like the following:
$$
cos(\omega t-k_xx-k_yy-k_zz)
$$
where $\omega = 2 \pi /T$, where T is the period of the wave, and $k_x,k_y,k_z$ are the component of a vector $\bf k$ whose intensity is $2 \pi / \lambda$ ($\lambda$ being the wavelength) and whose direction is the direction of the propagating plane wave.
What really matters, for the description of the wave, is the peculiar space-and-time dependence of the argument, not the alternation of $sin$ and $cos$ function when taking derivatives. 
Summarizing, there is no physical base for playing with other mathematical functions in the hope of finding something interesting. First of all because the special role played by $sin$ and $cos$ hinges on the Fourier's analysis. And also because the essence of the wave behavior is in the argument dependence on space and time.
