Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure? This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there.
$\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of $\Bbb{CP}^{2n+1}$ at one point.
On the other hand, $\Bbb{CP}^2 \mathbin{\#} \Bbb{CP}^2$ does not even support an almost complex structure: Noether's formula demands that its first Chern class $c_1^2 = 2\chi + 3\sigma = 14$, but if $c_1 = ax_1 + bx_2$ (where $x_1, x_2$ generate $H^2$, $x_1^2 = x_2^2$ is the positive generator of $H^4$, and $x_1x_2 = 0$), then $c_1^2 = a^2 + b^2$, and you cannot write $14$ as a sum of two squares.
Using a higher-dimensional facsimile of the same proof, I wrote down a proof here that $\Bbb{CP}^4 \mathbin{\#} \Bbb{CP}^4$ does not admit an almost complex structure. The computations using any similar argument would, no doubt, become absurd if I increased the dimension any more. 
Can any $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ support an almost complex structure? 
 A: The question has been answered in the crosspost to mathoverflow by Panagiotis Konstantis. I copy this answer here to close the question.

The $m$-fold connected sum $m\# {\mathbb{CP}}^{2n}$ admits an almost
  complex structure if and only if $m$ is odd, as we show in our recent
  [preprint][1]. (By the way, thanks to Mike for this interesting
  question, which motivated us to write the paper!)
Here's a brief summary of the proof's idea. Our main tool is a result
  by Sutherland resp. Thomas from the 60s which tells us when a stable
  almost complex structure is induced by an honest almost complex
  structure: this is the case iff its top Chern class equals the Euler
  class of the manifold.
As the connected sum of manifolds admitting a stable almost complex
  structure admits one as well, we certainly have stable almost complex
  structures on $m\# {\mathbb{CP}}^{2n}$ at our disposal, and we can
  understand the full set of all such structures by explicitly
  determining the kernel of the reduction map from complex to real
  K-theory. We then compute the top Chern class of all these structures:
  luckily for us, it turns out that in order to show the non-existence
  of almost complex structures for even $m$, it suffices to compute its
  value modulo 4 and compare it to the Euler characteristic of $m\#
> {\mathbb{CP}}^{2n}$. For odd $m$, we explicitly find a stable almost
  complex structure for which the criterion above is satisfied. 
[1]: https://arxiv.org/pdf/1710.05316.pdf

