My question is: is it at all possible for a single microorganism to develop resistance to everything we can possibly develop?
If you are only discussing antibiotics, the answer is, yes, it's theoretically possible but very highly unlikely, as different classes of bacteria have different characteristics that allow resistance to certain antibiotics to develop.
For example, if a bacteria doesn't have penicillin-binding proteins - allowing cross-linking and stabilization of the bacterial cell wall - then resistance to this antibiotic which makes susceptible bacteria unable to synthesize their cell walls - is meaningless, so unlikely to happen (except through sharing of useless genetic material, which does have limiting factors.)
If you're discussing all antimicrobials (and you consider something like bleach an antimicrobial), the answer is "no".
The danger isn't that a microbe will be resistant to every antibiotic. The danger is that it will be resistant to enough of them to cause a significant problem. It takes time to test for antibiotic sensitivity; in the meantime, a person with an infection that needs antibiotic treatment is getting sicker. Labs start testing for antibiotic sensitivity with antibiotics that the identified organism is usually sensitive to. If that fails to reveal sensitivity to an antibiotic at acceptable levels - one option is to increase the amount of antibiotic given - new tests need to be carried out, taking even more time (for antibiotic combinations or "atypical" antibiotics). It's possible that an antibiotic (or combination) will be ascertained only after that person's death.
Staph aureus has lived closely with man for I don't know how many hundreds of thousands of years. Most of the time, it's presence is benign. It's only when it becomes invasive that antibacterial resistance is a factor in dealing with it.
Antimicrobial Resistance Learning Site Although this is a veterinary site, it's clear and informative, and, though most people are unaware, use of antibiotics in feed animals - which falls under veterinary medicine - is a major contributor to development and spread of antibiotic resistance.