How many moment of inertia about center of mass exist? So imagine we have a rigid body and we want to find the moment of inertia about center of mass . Doesnt exist infinite axis that pass trough center of mass therefore infinte moment of inertia? Do they all have same value?
 A: I'm not sure how comfortable you are with vectors and matrices, but I'll offer this answer in case it helps.
No matter how complicated the rigid body is, in terms of its shape and the distribution of mass within it, the inertia can always be represented as a symmetric $3\times 3$ matrix
$$
\mathbf{I} = \begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ 
I_{xy} & I_{yy} & I_{yz} \\ I_{xz} & I_{yz} & I_{zz} 
\end{pmatrix}
$$
where I have assumed an $(x,y,z)$ set of Cartesian coordinates fixed to
the rigid body.
So, just six numbers are needed to tell us everything about the inertia.
By this, I mean that if you choose the rotation axis to be in the direction
of a unit vector $\mathbf{n}=(n_x,n_y,n_z)$ in the same coordinate system,
the moment of inertia about that axis can be written
$$
I_n = \mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n} 
= \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} I_{\alpha\beta} \, n_\alpha n_\beta
$$
As you say in your question, by changing the direction of $\mathbf{n}$,
you can continuously change the value of $I_n$.
Things can also be made simpler by choosing a set of $x,y,z$ axes that make $\mathbf{I}$ diagonal. This is always possible for such a matrix. In that case, the coordinate axes are called the principal axes of the rigid body, and the diagonal components of the matrix $I_{xx}$, $I_{yy}$, $I_{zz}$, are called the principal moments of inertia.
Some of this is covered on the Wikipedia page for Moment of Inertia, although it is a bit buried in complicated expressions.
The bottom line is that, although there is an infinite number of moments of inertia, if you know the three principal moments of inertia, and the principal axes that they correspond to, you can calculate the inertia about any axis.
A: The moment of inertia is always refered (and calculated) relative to a specific axis, so yes there is an infinite number of them, and in general they are different, unless you have some symmetry, like in the case of a sphere
A: Each rigid body regardless of shape can be represented by three principal moments of inertia that correspond to three orthogonal directions. 
If you do the math, you will see that you need two quantities to specify one of directions (like latitude/longitude), and an additional quantity for the second direction (spin about first direction). Once the two orthogonal directions are set the third one is calculated from their cross product. 
So, in the end, you need six quantities to fully describe the mass moment of inertia of any rigid body. 
Now because the above system is rather complex, typically MMOI is described with 6 quantities along known directions, typically aligned with the geometry. This is done with a 3×3 symmetric matrix, which has six independent quantities (three in the diagonal, and three in the off-diagonals). 
If you chop up a body into many small parts, each with mass $m_i$, and located at $(x_i,y_i,z_i)$ you find the mass moment of inertia matrix as
$$ \mathbf{I} = \sum_i m_i \left[ \matrix{ y_i^2+z_i^2 & -x_i y_i & -x_i z_i \\-x_i y_i & x_i^2+z_i^2 & -y_i z_i \\ -x_i z_i  & -y_i z_i & x_i^2+z_i^2} \right] $$
The 3×3 matrix transforms a 3×1 rotational velocity vector $\boldsymbol{\omega}$ to a 3×1 angular momentum vector $\boldsymbol{L}$
$$ \boldsymbol{L}_C = \mathbf{I}_C \boldsymbol{\omega} $$
Notice the designation $\square_C$ indicating the center of mass as a location.
Now to find the MMOI of an intermediate axis (not one of the chosen ones) you apply a unit rotation about the axis ($\boldsymbol{\omega} = 1\,\boldsymbol{n}$), and measure the angular momentum about the same axis, using a projection.
$$ I_{\text{axis }\boldsymbol{n}} = \boldsymbol{n} \cdot \left( \mathbf{I} \boldsymbol{\omega} \right) = \left( \boldsymbol{n}^\intercal \mathbf{I} \boldsymbol{n} \right)$$
Since there are infinite directions, here are infinite numeric values that can be produced by this calculation, but they are always going to be a combination of the three principal moments.
For example, the MMOI about the z-axis might be
$$ I_{\varphi} = I_{yy} + \cos^2 \varphi (I_{zz}-I_{yy}) $$
