On this thread, the comment that I had written up, but couldn't post once the thread was deleted, is too good not to post somewhere. So here it is.
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You start with a sun at the center, and start adding air near it. You want to get it to the point, let's say, where at the current distance of Mars' orbit, the air pressure is one atmosphere. As DU noted, it's going to be much higher near the surface of the sun, and just because it's 1 atm at 228 million miles doesn't mean there's a sudden cutoff and beyond that it's zero - it gradually dissipates at greater distances beyond that. Let's also assume that at the start you're at least adding it quickly enough so some of it sticks around; it doesn't all escape as fast as you add it.
Here's the rub: soon, you've added enough air that as you add more, you have to take into account not just the gravity of the sun, but the gravitational effect of the air itself. The air will compress itself closer and closer to the sun, not just due to the gravitational effect of the sun, but the gravitational effect of the air itself (which, we'll assume, is centered on the sun). I strongly suspect that you'll add enough mass to form a black hole before you've added enough mass to generate 1atm pressure in that large of a volume.
Also, pressure is dependent on temperature, so you'd also have to be able to figure out just hot that air is getting in order to generate the required pressure.
But we may be able to evaluate the "which happens first: 1 atm or black hole?" question. Let's make some wild, completely unwarranted assumptions, such that the gas is at 300K (27°C), that, in spite of gravitational effects, it's uniformly distributed throughout a sphere with a radius of 228 million miles (and ignoring any gas beyond that), and that the air in question is molecular nitrogen. This would require around
2*10^35 kg of nitrogen.
To create a black hole with a radius of 228 million miles requires about
2*10^38 kg. So it's hard to say. If you could uniformly distribute the mass within the sphere, you could do it without creating a black hole, but given that gravity will compress the air more towards the center of the sphere I think you're still going to end up with a black hole.