I merely qualify my "as far as I am aware only humans have bothered to measure it" statement because for all I know some sentient race 'out there' in the vastness of the universe may exist who take time as seriously as we do.
As for trying to connect time to infinity, I'm sorry but you are barking up the wrong tree. Time is one thing, infinity another, the former expresses the progression of events around us the latter something not just big but eternal which goes beyond the universe and time and everything but which is impossible to achieve.
Its a bit like zero which in ancient times did not exist and everything started with number one (its why we are in the twenty first century when it clearly says 2012 as their was no year zero) as the idea of nothing seemed impossible, which is actually true but mathematicians soon began to realise that without zero maths gets really complicated so it was in effect 'invented' to make things simpler. Around the year 130 ad Ptolemy used Omicron as a substitute for zero but it wasn't until the seventh century ad when it began to be used in India. So, once you have a concept of nothing then you start to think about the possibility of everything and so you get infinity. But its not real, ok?