There are very few jobs in first world that do not do background checks. However some companies do not do background checks for junior level stuff and generally it's only when one moves to high level position, they may do background checks.
Specific industries include food industry – restaurants, fast food like MacDonalds, junior office staff, adult film star, ware houses, construction.

Some employers only do count or state checks for which they are in. Most employers are interested in 5 or less years offences. Very few can hold anything that is older than 7 or 10 years depending on the employer or the offense.

Generally, people who are worried about background checks have something to hide. You may need some professional help to offer some suggestions on how to address it in your job search. I would suggest you research those organizations that may specialize in your particular issue, such as the “Safer Foundation”, and seek help directly from them.