About as close to zero as you might consider. 0.03% is the humidity at the South Pole.
We regularly get humidity in the single digits where I live in Australia. Temperatures in the low 40’s (Celcius) to match. I’m sure Arizona is pretty similar.
So what combination of heat and humidity is an imminent threat to life?
This Canadian site has an interesting table about that. They calculate a “humidex” and a score of over 54 means imminent risk of heat stroke (which untreated means death) for someone doing a moderate level of work.
According to that site at a temperature of 100 F (37.8 C), heat stroke would be an imminent risk at a relative humidity of about 65%.
At a relative humidity of near 100% (call it 95%) a temperature of 95 F (35 C) would also mean that imminent risk of heat stroke.
So the op’s point is valid but it doesn’t take 100/100 to do it.
I was going to suggest that a hot bath would be the physiological equivalent of 100/100, but then this site says:
I wonder how many people clicked on the link just to say this (I lived in Houston from 1971-1986 and lived on the coast without AC before that.)
It is if the environment is over 98 degrees…
As far as Houston goes … yeah it’s subtropical and it gets hot but not into heat index 130 (= humidex 54 = risk of heat stroke) range hardly ever. The record hot day had a temperature of 104 and an extremely uncomfortable heat index of 110 … but not 130.
And isn’t there more air conditioning per capita in Houston than any other city in the world? Who goes outside? 
Having covered the south central U.S., how about some place really dismal, like a tropical rain forest? Let’s say it’s a hot summer day in a near-sea level swamp. How high does the heat index get in a place like that?
I found China to be really hot and humid when I was there last summer, Hangzhou in particular. So, I check Weather Underground for today’s forecast.
93 F, with humidity of 73% predicted at 5pm (their time) gave a heat index of 120 F.
Pfft. I used to spend summers in Muscat, Oman (summer temperatures occasionally exceed 120 degrees Fahrenheit) and one year we had 110+ degrees and 80+ per cent humidity for five days in a row.
God, that was miserable.
Living in Houston, I agree that it is hot and miserable in the summer, especially without air conditioning (the week without power during Hurricane Ike was excruciatingly unpleasant). Today it’s supposed to get near 100, but the humidity will probably not exceed 50% when it reaches 100. As Darth Panda and RNATB have posted, the weather gets a whole lot worse in places closer to the tropics.
One of my favorite weather anecdotes came from an American professor visiting Vietnam. He was fluent in Vietnamese, but looked like your basic U.S. tourist. He said it was about 100 and steamy when he got off the plane, and so by the time he made it out of the terminal, he was drenched. A little girl walking with her mother tugged on mom’s hem, pointed at him, and asked her mother, “Mommy, why is that man melting?”
This may be a nitpick, but the hottest temperature ever recorded in Houston was 109 degrees, on September 4th, 2000. The dew point was recorded at 66 degrees F, giving a heat index of 117 degrees (and this is where anyone from Phoenix can start laughing.)
Keep in mind that the heat index from hangzhou that I posted (120) wasn’t some kind of record or anything - it’s just what the forecast happened to be on that day…
From what I can find, the heat index on our hottest day (122F, 6/26/90) was 134F. That was in June, though, our dryest month. I bet some of our July days of 115+ have had much higher humidity and much higher heat indexes.
Back when I was training on a prototype naval nuclear power plant (D1G), there was a so-called “steam room” in which there were three main steam valves that had to be manually opened and shut when the plant was started up and shut down.
We were issued thick leather gloves (with government markings on them reading, “Gloves, Leather, Forest Worker” :)) to operate the handwheels, because they were hot (over 200 deg F). It took about five minutes to open and shut each of the three valves, and it was hard work.
Anyway, there was a prominent thermometer in the room right next to a wet-bulb thermometer. Both generally read the same reading, indicating the relative humidity in the room was 100%. And both generally read a temperature of about 125 deg F.
As an “Engineering Officer of the Watch” in training, I only had to do this operation twice (once for a startup and once for a shutdown). When I left the room, my undershirt and cotton uniform were literally dripping with sweat. I looked like a steamed lobster. I actually got salt stains all over my uniform from all of the sweat.
So anyway, a human can survive these conditions, for a little while, anyway. 
P.S. Once I got to my submarine, we had nuclear-powered air conditioning throughout the engine room–except during drills when we had to “rig for reduced electrical.” With the A/C out, temperatures in the engine room rose from 85 deg F to 120 deg F (at close to 100% humidity) in about 15 minutes.