Why is the number of PCR cycles limited?
I would draw the line beyond 35, but thats a bit cosmetic. 
The reasons are manyfold:


due to the exponential fashion of the amplification (ideally)
reagents are used up at some point
reagents degrade, this is especially true for the dNTPs
the activity of the enzyme, despite being heat-stable is declining
over time
beyond 35 cycles the exponential curve is flattening out (reasons see
above)
if you run the PCR for too long, you will get more and more
side-products (mostly primer dimers, but mis-aligned primers can also
make problems), this is more a problem for real-time PCR than for ones run on a gel


If you need a higher sensitivity with more cycles, you can use the technique called "nested PCR". There you do a first round with a primer pair specific for the region of interest and then do a second round with primers which are located slightly to the inside of the amplified DNA. This is done to avoid the amplification of unwanted contaminations. Since you do some 50-70 rounds of PCR amplification in total, this method is extremely sensitive (also to contaminations). See the image from the Wikipedia article for details: