Cosmology is an interesting science. It is a combination of astrophysics and history because every time you peer at the farthest objects in the Universe, you are looking back in time. By studying the objects we can see now we can also determine what they were like before we could have viewed them. Unfortunately, even though the passage of time means we can view more of the Universe, our ability to figure out what everything was like pre-observation is being diminished by more recent interactions. Researchers at Harvard-Smithsonian Center for Astrophysics decided to crunch the numbers and find when the optimal time for viewing the Universe would be.
The answer, unsurprisingly, is in the distant past when the Universe was only a half billion years old. This is the same time that the first stars and galaxies were forming, which is not a coincidence. Stars and galaxies are massive structures and their formation requires the original structure of the Universe, left by the Big Bang, be destroyed. It is like the reverse of Humpty Dumpty, in that the pieces once made into the egg, cannot be broken the same way again.
Though the optimal time to observe the Universe has passed, it is not impossible to study the prehistoric Universe. Radio observatories can explore the 21 cm emissions from 13 billion year old hydrogen gas to learn what regular optical observatories cannot. However, there will still come a time when objects are so far away, and the Universe is expanding so fast, that light from those objects will be unable to reach us, ever.