Can Fuel Exhaust Be Recycled
One of the major components of exhaust is carbon dioxide, produced by the combustion of hydrocarbons - if in complete combustion, follows the general reaction:

$$\ce{C_{x}H_{y} + O2 -&gt;CO2 + H2O}$$

Other possible emissions include $\ce{NO_{x}}$, particulate matter, carbon monoxide, VOCs etc. The amounts of these emissions vary from car to car (due to a lot of factors, e.g. age, wear-and-tear etc). According to a self study by Audi, in the document Motor Vehicle Exhaust Emissions, on page 6, they determined that asides from $\ce{N2}$, the main emissions were $\ce{CO2}$ and $\ce{H2O}$.

For this answer, the focus will be on the possibility of recycling the $\ce{CO2}$ emissions (and linking to the $\ce{H2O}$ emissions).

This very question is the subject of the Scientific American article Reverse Combustion: Can $\ce{CO2}$ Be Turned Back into Fuel?, the articles details that there are several research groups looking at this problem. 

A couple of examples of ideas and experiments that have been put forward include:


  In the 1990s a graduate student named Lin Chao at Princeton University decided to bubble carbon dioxide into an electrochemical cell. Using cathodes made from the element palladium and a catalyst known as pyridinium—a garden variety organic chemical that is a by-product of oil refining—he discovered that applying an electric current would assemble methanol from the $\ce{CO2}$.


Inspired by the natural process that removes $\ce{CO2}$ from the atmosphere - photosynthesis, continued research has sought to emulate this - essentially, the current research is effectively looking at a possible solution by taking


  $\ce{CO2}$, water, sunlight and an appropriate catalyst and generate an alcoholic fuel


The research reported in the Scientific American article and the National Geographic article Carbon Recycling: Mining the Air for Fuel go along the lines of extracting the excess $\ce{CO2}$ from the air rather than directly from the car's engine or exhaust. The reason for this is the current practicality, with the National Geographic reporting that


  The Sandia prototype’s solar collector has an area of about 20 square meters (215 square feet) for a reactor the size of a beer keg, Stechel says. About 300,000 acres (121,400 hectares) of mirrors would be required to collect enough sunshine to make the equivalent of 1 million barrels of oil per day, she says. (The world currently consumes about 86 million barrels per day of petroleum and other liquid fuels, including biofuels.)


The research shows that, yes it is theoretically possible and practically possible on a larger scale to recycle exhaust gases.