Is a plane wave necessarily monochromatic? 
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*Is the expression $$\psi(x,t)=A\exp{i(kx-\omega t)}, \hspace{0.3cm}A=\rm constant$$ the most general form of the plane wave? 

*If yes, does it mean that a plane wave is necessarily monochromatic? 

*If not, what is an example of a monochromatic wave that is not a plane wave? What is an example plane wave that is not monochromatic?

*If the answer of (1) is in negative, what is the most general form of a plane wave?
 A: Strictly speaking, a plane wave is a wave with a planar phase front, which implies more than one spatial dimension. Typically, we would speak about plane waves in three dimensions. It also implies that the phase front is well defined, which requires it to be monochromatic. In that case the expression you provide is not the most general expression for a plane wave. Instead one would express it as 
$$\psi({\bf x},t)=\exp(i\omega t - i{\bf k}\cdot{\bf x}) , $$
where ${\bf k}$ is the propagation vector.
As such it is monochromatic. I can imagine that one can compose a wave consisting of a spectrum of such monochromatic plane waves, all propagating in the same direction, but wiht different frequencies. However, whether that would still represent a plane wave is not certain, because the phase may be ill-defined for such a polychromatic wave.
One can construct various types of waves that are monochromatic, but not plane waves. The only condition is that they have a fixed frequency. All such waves can be composed as a superposition of plane wave all having the same frequency. One example is the Gaussian beam, which is a solution of the paraxial wave equation
$$ \nabla_{xy}^2 \psi({\bf x}) + i 2 k \partial_z \psi({\bf x}) = 0 . $$
Note that the equation contains the wave number $k=\omega/c$ as a parameter, which means that for a specific value of this parameter, the frequency is fixed. Hence, solutions of this paraxial wave equation are monochromatic fields.
