This is one of those questions that produces more questions than answers.
First is, there is no hard answer as to a safe depth on air. George I. of the WKPP considers air to be unsafe at any depth. Hmm, sounds like a title for a book.
There are lots of factors to consider;
1. Who is diving? training, skill, experience, etc.
2. Why are they diving? look around? collect something? work?
3. What surface support and backup is available? support divers, rescue facilities, location of chamber, etc.
4. SCUBA of Surface Supplied? Surface supplied with a com link or scuba with a wireless com link allows better monitoring of the diver and provides a greater margin of safety.
Now on to some more direct answers.
I have done lots of dives to 130 feet on air but I don't dive that deep anymore on air. I also used to drink alcohol but I don't do that anymore either.
On scuba I will not go deeper than 100 feet on air. I prefer to stay above 70 feet on air.
Nitrox is good to the depth limit for the O2 content. I prefer to use 1.2 ATM PPO2 maximum for diving with a standard regulator and 1.4 if using a full face mask. The use of 1.6 ATM PPO2 is for decompression ONLY and ONLY when resting. O2 tox is one thing that can kill you very quickly and easily. Not that the O2 kills you but the unconsciousness and drowning at depth makes for a very complicated rescue/resusitation. This is why I will allow a higher PPO2 with the FFM, it keeps the divers face dry and supplied with gas so that when you body starts breathing again after the convulsions you breath gas and not water.
For deeper than 100 feet mixed gas with He is the way to go. This requires much greater training and different deco profiles than air. It also requires a much higher standard of care in planning and checking dive gear and procedures before each dive.
I would consider diving to 165' on air only if surface supplied, with a com link, safety diver fully outfitted and ready to go and a deco chamber on site. Then again, that is not SCUBA.