For the longest time I felt guilty about getting a daily newspaper (this was also the time I thought my community didn't recycle newspaper, where I go that I don't know...)
Good thing I started gardening soon afterward because newspaper makes for excellent use in the garden. I was worried about potential ink chemicals when I started, but after research I found that those worrisome inks have been replaced by soy based one. I still take out the glossy paper though for the recycling bin as I am always wary of those coatings they use.
It takes a few weeks to build up a really good pile, but I find it most excellent keeping the paths around my garden beds weed free, usually only able to do one side of the side at a time. I apply 10-15 layers of newspaper down and then add a good thick pile of dried out grass clippings I gather around the neighborhood and stomp them together. A bit of rain makes it flatten and stick real well, but if it gets very soggy, you may want to reapply. This path method is especially good as I have a tendency to toss my weedings into the paths, but all the stomping and smothering them with newspaper keeps them at bay.
Newspaper is also a wonderful, inexpensive mulch. Just shred it up as fine as you like and apply to your beds. Water goes through it and retains moisture quite well too. The earthworms take it down eventually or it just breaks down into the soil as a amendment. It also makes for some fine reading as you garden :)
Another neat trick is to loosely roll up some newspaper and rubber band it in the middle so that there are lots of little nooks and crannies. Wet the paper bundle thoroughly and leave it out by a pest ridden area of the garden. Come back early in the morning your newspaper trap will have slugs, sometimes problematic woodlice, and other pests will have congregated there and you can either toss the entire batch of them with the paper or pick them off.
Also worth mentioning, though I haven't done yet because I have trays, is that newspaper can be made into inexpensive starter pots. Take a large can and a smaller one to fit within it, and place a few sheets of open newsprint on top of the mouth of the large can, then fit the smaller can within to shape the newsprint into a can shape! Your paper pot is complete and just fill with appropriate soil and seed and voila, ready to grow!
My favorite use for newspaper in the garden however is for composting. I have difficulties finding good "browns" for my bin (few branches/twigs, plus no good method to chop them up here) and I don't want to use the neighbors' potentially treated grass clippings in my compost that I put in the garden, so I shred up newspaper and add it to the large indoor compost bucket to soak up any smells and extraneous moisture before I add the entire thing to the large composter outside when I have the time, adding more newspaper at that trip.
May all your use of newspaper be put to good use! Let's give our trees some dignity!