Your problem is a very old idea and already known since at least ancient Greece.
Please take a look at this wiki page on Zeno's Paradoxes:
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
The problem has physical limitations in that you will have to stop dividing when you come to the scale of atoms or molecules.
If you disregard that you will be limited by what can be observed which is at least 10 magnitudes larger than the smallest scale.
Ultimately you will be limited by the smallest scale science currently thinks exists fundamentally, the Planck scale or 1.616252×10−35 m.
More info here:
http://en.wikipedia.org/wiki/Planck_scale
If you go even further and disregard all practical and physical limits you will encounter logical problems.
At present the majority of logicians, philosophers and mathematicians regard infinities outside of concepts in math like for example in set theory as impossible.
The distinction needs to be made between conceptual infinities and real infinities.
Real infinities are officially termed actual infinities:
http://en.wikipedia.org/wiki/Actual_infinity
All this would not stop you from considering the future as an infinity.
Such infinities however are not actualized yet so they are called a potential infinity.
Some people consider God an actual infinity (also called the absolute infinity) while others consider the infinity related attributes of God proof for His non-existence.
Some links related to infinity that may be of interest that might keep you busy for a while:
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
http://en.wikipedia.org/wiki/Supertask
http://en.wikipedia.org/wiki/Balls_and_vase_problem
http://en.wikipedia.org/wiki/Finitism
http://en.wikipedia.org/wiki/Infinitesimal
http://en.wikipedia.org/wiki/Hyperreal_number
http://en.wikipedia.org/wiki/Surreal_number
http://en.wikipedia.org/wiki/Transfinite
http://en.wikipedia.org/wiki/Galileo%27s_paradox
http://en.wikipedia.org/wiki/Cantor%27s_paradox
http://en.wikipedia.org/wiki/Burali-Forti_paradox
http://en.wikipedia.org/wiki/0.999...
http://en.wikipedia.org/wiki/Aleph_number
http://en.wikipedia.org/wiki/Axiom_of_limitation_of_size
http://en.wikipedia.org/wiki/Dedekind-infinite_set
http://en.wikipedia.org/wiki/Euclid%27s_theorem
http://en.wikipedia.org/wiki/Russell%27s_paradox
http://en.wikipedia.org/wiki/Skolem%27s_paradox
Note that nature is considered fully consistent. That is the basis of all knowledge.
If a paradox exists it has to be caused by problems in man-made concepts (or axioms in mathematics).
This means paradoxes must be analyzed by trying to find the problem in the definitions or concepts.
The chicken and the egg problem for example is caused by the concept chicken.
The concept chicken is something practical for humans but in nature lifeforms exist in a constantly changing form.
Through many steps of gradual change transitions are made so there never is an exact point where the first chicken appears, the bird just became more and more chicken like.
The paradox appears when we create a concept that sets an artificial boundary called chicken.