I'm not really seeing a lot of truly free things that cost something now. Maybe because I'm a youngin?
Free air - No need for me. I bought a $6 portable 12v DC air compressor and a $1 air gauge to top off the tires to the specified PSI as needed. It also is available everywhere my car is because I keep it in the trunk. Obviously this obviates the need for free air.
Free diagnostics - I see signs on shops that offer this, but I'll bet you a shiny nickel they find something that needs fixing that they are also running a special on.
My Honda dealer does the free 35 point courtesy inspection with their $30 oil change or any service really. They always find something critically wrong with the car that they can fix for $100-200 and do it quickly. 90% of the time, after I research the issue, I determine the repair isn't needed and their recommendation is borderline fraud. But they are cheaper than the local mechanic that is slightly more trustworthy, as long as you don't fall for the revenue enhancer services they tell you that you critically need. "Free" is expensive sometimes.
I do continue to get great free service at Discount Tires (America's Tires in some regions apparently). Free rotations and balance and flat repair and air refills for the life of the tires if you pay them to install your tires. That one has actually saved me money over the years. But they try to up sell about half the time I bring the cars in for a rotation and balance. It is always something - warped wheel, (barely perceptible) uneven wear, special on tires this week only, we can't rotate because they are worn too much, etc. Usually I can talk them down and still get the "free" service that was part of the inducement for me to continue patronizing their business year after year. It is obvious that they offer the free service to keep you loyal and to bring you in for a valuable customer interaction in store with serious information asymmetry for almost all customers. I mean how can you NOT spend $400 on a new tire and wheel for your wife's minivan when they tell you how bad the condition is on one tire and that if it failed your wife will indubitably crash and burn and die with all your kids in the van?
Sometimes they are right, but you have to know when they are just trying to convert a free visit to a sale.
Libraries are great, and ours keeps getting better. They just remodeled ours in the neighborhood (grand opening this weekend!), they just upgraded the online reservation system to make it more e-commercy (with a shopping cart feature). Online e-books check out is available. Tons of fast computers and free wifi. Free kids programs.
I have also noticed a trend lately of businesses sending out totally free coupons. In the last week we received a $10 free coupon to a department store, totally free meal for one to a national chain buffet restaurant, 3 coupons totaling $30 off of $30.03 in purchases at a big box hardware store, and 2 free food items from a fast food restaurant (although they were from something purchased). I'm not even sure I'll use all these free things.