Question: When I consider my own existence with respect to time I can imagine three possibilities: 

(1) **Time extends infinitely into the past.** In this case, how can the present, with me in it, exist, since there would be an infinite period of time that existed before the present. That seems illogical. 

(2) **Time had a starting point from which everything in our universe evolved to the present (t=1).** That possibility would require something (God) to start the clock ticking (t=0). But in this case we're back to square 1 with the problem of an infinite past. So the only possibility that makes sense to me is a third possibility 

(3) **Where time is some sort of abstraction in human consciousness that we need in order to relate cause and effect.**

Answer: 
Time as you perceive it is naturally a mental construct, but it seems to correspond to an element of reality.

"In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system" - [EPR](http://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777)

However, we know reality is absurd

"The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. And it fully agrees with experiment. So I hope you can accept Nature as She is - absurd." - [Feynman](http://www.abebooks.com/book-search/title/qed-the-strange-theory-of-light-and-matter/first-edition/)

So there you have it, a fourth possibility that time corresponds to an element of reality and absurdly continues infinitely into the past.

Other people here proposed to view time as the domain of integers or as an axis of real numbers, but how is causality represented in such an analogy?
(1) is a little too loaded or existential of a question. If we shoot a classical arrow through the air, we can slice up each moment of its flight time into infinitely many infantecimal moments. In a sense, time extends infinitely into the present moment. What is "continuous" motion but the mathematical model of the rational number? In one second, there are an uncountable number of fractional sub-seconds; to say it another way, you can split 1.000...0 into an infinite number of fractions.

The concept of "infinite" is an abstract mental construct. By very definition, "infiniteness" cannot be measured. By that admission, it might be tempting to conclude it is a fabrication. It might be an idea that only exists inside the human mind. Another one of our 'constructs' to provide order and explanation to what we describe as "uncountable" processes and rules of logic.

That doesn't mean it doesn't exist or isn't correct, there's more than enough scholarly work to suggest it is a natural and 'real' thing either.