I just realized I never relayed this weird incident: I've been trying not to buy things, but a couple of months ago I found something I've been after for a long time: a dark gray chambray shirt that's not a work shirt but styled like a casual shirt with a button-down collar and a single pocket. It was at Old Navy and was only $16, so I bought it online, along with another item from Gap.
When the shirt arrived, I immediately noticed it had a weird smell, not necessarily a bad smell, but something you shouldn't smell from a new garment. It was a chemical smell, or possibly petroleum-based; I could never pin it down definitively. The Mrs. said to her it smelled like mildew, but my nose wasn't getting that, and there was no visible evidence to support either conclusion. I ran it through the wash and hung it up to dry. I went downstairs the next day to get it, and the smell was still there, so a couple of days later I washed it again with no change.
I left it hanging in the basement for a while and periodically checked it, but the smell remained. I thought perhaps our scent-free detergent was not strong enough to eradicate the smell, so a couple of weeks later I took a walk to a nearby laundromat and bought a single-load package of Tide from the vending machine. Tide's scent is so strong I figured it could take care of anything. Same process, same result: as the Tide smell faded away, I could tell the original smell was still there, though it did seem like it may have been a little less noticeable.
I had used only about half of the Tide, so I ran more water in the washing machine, added the detergent, and let the shirt soak for several hours before running another wash cycle. When I hung it up all I could smell was Tide. I stayed away from it for a couple of days, but by then the Tide had faded away again, and there was no change in the shirt's smell. I reasoned that it had to be a component of the dye, or some other aspect of the production process.
This was much more effort than I had ever had to make to remove a smell from clothing, and I decided I had done enough. As much as I wanted to keep the shirt, I had no choice but to return it. Curiously, it had disappeared from the Old Navy site around the time my order arrived, so I was unable to return it for an exchange. At the store a clerk checked the inventory and found two in the same size, but in New York state somewhere. It wasn't worth it to me to pursue another one. I think I just wasn't meant to have that particular shirt, and so the search continues...
Monday, 7/14: After I got a comment on this post, I realized that I had left out part of the story. Two people suggested soaking the shirt in a baking soda and water mixture, so after all the washings did not help, I did soak the shirt, but unfortunately that effort did not help either.