You should not confuse the terms light years $[ly]$ to be a unit of time, it is actually a unit of distance, the distance light will travel in vacuum in one year.
When we speak of the age of the universe we use years, in particular about $13.7$ billion years.
Now do to the expansion of the universe the distance form which the light comes to our eyes or telescopes can come from region further away then simply $13.7$ billion $ly$. If you do the calculation you obtain a result for the radius of about $45-47$ billion $ly$.
Answering your question, if you look at a distance of about $13.7$ billion $ly$ with your telescope you will simply see a region of space younger than $13.7$ billion years which is nothing special.
Maybe what you are really asking is what we will see if we look for the regions that are the oldest observable one. Well we actually have a picture coming from that period and is the CMB (cosmic microwave background).
This is the oldest light that we can detect, is from when the nucleosynthesis took place, at that time the universe became transparent.
Before that period the only hope to get some information come from neutrino or gravitational waves detection. This is also one of the many reasons because neutrino and gravitational waves detection are so interesting. They can give use new information about the birth or our universe. The only problem is that we have never observed gravitational waves directly (there are proofs of their existence from energy loss from binary systems), and that neutrinos are really hard to detect.