Slime is a gimmick and does not work in high pressure tires and only slows small leaks in lower pressure ones. Slime patches do work ok.
A puncture proof tire is a myth, if one really existed you would not want to ride it anyway it would be way to heavy and slow. In a desparate moment, I put a Specalized all condition pro on my rear wheel a month ago. Since then I have had 3 flats (a very small needle shaped piece of wire was embedded in the tread) and the tire is wore all the way to the threads after only about 500 miles.
I do ride with thorn resistant tubes and run of the mill 700x23 tires in lesser road bike. I cant recall ever having a flat with those tubes in, but they do make the bike feel slower.
My reccomendation is to be prepared. Leave a few minutes earlier. Get decent tires and learn to practice fixing a flat. The proper tools are inexpensive and a tire should be able to be repaired in a matter of minutes. After a ride or riding an especially debris filled section, inspect the tire and remove any embedded fragments. Be aware of the path ahead of you and avoid debis when you can. I used to get a flat every week or so, with experience it seems that I may get one or two - on the road - flats a season.
In the end, you would be more delayed by foul wind / weather than you should be by a flat tire.