I was looking through my bathroom window this night and I was wondering if any of the photons my retina is hit with are from 13 (40) billion light years away ?!
I was looking through it a few seconds, was that too less of a time?
Do I need a Hubble for this?
There are two problems with seeing things a long way away. Firstly there's the sheer distance - the brightness of a star falls off as the inverse square of distance so after 13 billion light years anything is going to be pretty dim. Secondly there's the red shift. At 13 billion light years the red shift is about $z = 7$ i.e. the wavelength we see is a factor of eight $(z + 1)$ longer than the light originally emitted by the star. The longest wavelength we can see is about 700nm, so to see the star it must be emitting at a wavelength of less than about 100nm, which is in the hard ultra-violet.
We can do a rough calculation using the distant galaxy z8_GND_5296 as an example. I don't know the spectrum of z8_GND_5296, but we can do a rough calculation. Wikipedia gives the apparent magnitude as 25.6, and the Sun has an apparent magnitude of –26.74, so there are 52.3 orders of magnitude between the two. One magnitude is a brightness difference of $100^{1/5} \approx 2.512$, so 52.3 magnitudes is a factor of about $10^{21}$ i.e. your eye receives $10^{21}$ fewer photons from z8_GND_5296 than it does from the Sun.
A quick back of the envelope calculation tells me that the number of photons from the Sun hitting the Earth's surface is about $3 \times 10^{21}$ per square metre per second, so the factor of $10^{21}$ neatly cancels out and we conclude that the number of photons from z8_GND_5296 hitting the Earth is about 3 per square metre per second. This is a very approximate calculation, so you should regard this as an order of magnitude estimate at best.
According to Wikipedia the diameter of the pupil in your eye is around 9mm (assuming it's dark), and this is about $8 \times 10^{-5} m^2$, so the number of photons entering one eye from z8_GND_5296 is around 0.00025 per second. You would have to watch z8_GND_5296 for about 4,000 seconds for a single photon to enter your eye.
This isn't quite what you asked of course, because z8_GND_5296 is just one galaxy out of many at a distance of around 13 billion light years. I have absolutely no idea how many such galaxies you are looking at when you stare out of you bathroom window but there must be at least 4,000 and probably vastly more than this, so I'd say it's a fair bet that every second at least one photon and probably many more enter your eye from galaxies 13 billion lioght years away.
A non-negligible, say 1%, of the radio frequency noise in the natural environment is due to the cosmic microwave background radiation; I guess that some of those photons could "hit your retina"; but you can'really see them. There is a non-zero, although much much smaller, flux of optical frequency photons (TODO: estimate the flux of optical photons).
Technically you are seeing photons from the recombination phase, which occurred $\approx 377,000$ years after the big bang.
