This is a very good question, and I would really love to know the answer since its current state seems to be quite obscure. Below is just a collection of remarks, surelly not the full answer by any means. I would like to argue that for the moment there is no any deep mathematical reason to think that the Euler number of CY 3-folds is bounded. I don't belive either that there is any physical intuition on this matter. But there is some imperical information, and I will describe it now, starting by speaking about "how many topological types of CY manioflds we know for the moment". 


As far as I understand for today the construction of Calabi-Yau 3-folds, that brought by far the largest amount of examples is the construction of Batyrev. He starts with a reflexive polytope in dimesnion 4, takes the corresponding toric 4-fold, takes a generic anti-canonical section and obtains this way a Calabi-Yau orbifold. There is always a crepant resolution. So you get a smooth Calabi-Yau. Reflexive polytopes in dimension 4 are classified the number is 473,800,776. I guess, this number let Miles Ried to say 
in his article  "Updates on 3-folds" in 2002 http://arxiv.org/PS_cache/math/pdf/0206/0206157v3.pdf , page 519  

"This gives some 500,000,000 families of CY 3-folds, so much more impressive
than a mere infinity (see the website http://tph16.tuwien.ac.at/~kreuzer/CY/). 
There are certainly many more; I believe there are infinitely many families, but the contrary opinion is widespread"

A problem with the number 500,000,000 in this phrase is that it seems more related to the number of CY orbifolds, rather than to the number of CY manifolds obtained by resolving them. Namelly, the singularities that appear in these CY orbifolds can be quite involved and they have a lot of resolutions (I guess at least thousands sometimes), so the meaning of 500,000,000 is not very clear here.  

This summer I asked Maximillian Kruzer (one of the persons who actually got this number 473,800,776 of polytopes), a question similar to what you ask here. And he said that he can garantie that there exist at least 30108 topological types of CY 3-folds. Why? Because for all these examples you can caluculate Hodge numbers $h^{1,1}$ and $h^{2,1}$, and you get 30108 different values. Much less that 473,800,776. As for more refined topological invariants (like mulitplication in cohomology) according to him, this was not really studied, so unfortunatelly 30108 seems to be the maximal nubmer garantied for today. But I would really love to know that I am making a mistake here, and there is some other information.

Now, it seems to me that the reason, that some people say, that the Euler charachteristics of CY 3-folds could be bounded is purely imperical. Namely, the search for CY 3-folds is going for 20 years already. Since then a lot of new families were found.  We know that mirror symmetry strated with this symmetric table of numbers "($h^{1,1}, h^{2,1})$", and the curious fact is that, according to Maximillian, what happened to this table in 20 years -- it has not got any wider in 20 years, it just got denser. The famous picture can be found on page 9 of the following notes of Dominic Joyce http://people.maths.ox.ac.uk/~joyce/SympGeom2009/SGlect13+14.pdf . So, this means that we do find new families of CY manifolds,   all the time. But the values of their Hodge numbers for some reason stay in the same region. Of course this could easily mean that  we are just lacking a good construction. 

Final remark is that in the first version of this question it was proposed to consider  complex analitic manifolds with $c_1=0$. If we don't impose condition of been Kahler, then already in complex dimesnion 2 there are infinite number of topological types, given by Kodaira surfaces, they are elliptic bundles over elliptic curve. In complex dimension 3
Tian have shown that  for every $n>1$ there is a holomorphic structure on the connected sum of n copies of $S^3\times S^3$, with a non-vanishing holomorphic form. Surelly these manifolds are non-Kahler. So if you want to speak about any finiteness, you need to discuss say, Kahler 3-folds with non-vanishing holomorphic volume form, but not all complex analityc ones.