According to Grok,
Here are the countries most frequently cited as not requiring a criminal record check for standard residency pathways (based on multiple expat and immigration sources as of 2025–2026). These typically apply to options like temporary residency by financial means, retirement, or investment:

• Mexico — Temporary residency (leading to permanent) generally does not require an FBI report or police certificate. Focus is on proof of income/savings or property. Mexico is one of the most consistently mentioned "no background check" options.
• Chile — Background checks are not required for many residency applications (e.g., via income or investment). It’s often highlighted for its stability and infrastructure.
• Colombia — Temporary residency visas (which can lead to permanent after 5 years) usually skip the criminal record requirement. One of the easier processes in South America.
• Peru — Certain pathways (especially applying before entry via a representative) can avoid the need for a home-country police certificate or Interpol check.
• France — Some long-stay visas and residence permits (e.g., Visitor Visa or certain talent/passive income routes) typically do not require a police report from your home country.
• Italy — The Elective Residency Visa (Residenza Elettiva) for those with passive income often does not mandate a criminal background check.
• Cambodia — Business or extension visas (renewable long-term) are very straightforward and generally do not require a criminal record check. It’s often described as one of the easiest places overall.

Other Mentions (With Caveats)

• Argentina and Bolivia — In some cases, especially applying from inside the country or via territorial presence, they may only require an affidavit or local Interpol check rather than a full home-country police certificate. Enforcement can vary by office.
• Hungary (White Card digital nomad visa) and Mauritius — No criminal background check required for these specific temporary programs.
• Georgia — Occasionally mentioned for simpler residency routes without a full check.

Important Realities and Warnings

• “Do not ask” ≠ guaranteed approval — Even if no police certificate is required, authorities can still deny entry or residency if they discover serious criminal issues (e.g., via their own checks, Interpol, or at the border). Minor offenses are often overlooked in these lenient countries, but violent, drug-related, or immigration-related crimes can still cause problems.
• It depends on the visa type — Temporary residency or digital nomad extensions are more likely to skip checks than permanent residency or citizenship pathways. Requirements can also vary by consulate, application method (inside vs. outside the country), or changes in policy.
• Most countries do require it — Strict nations like the US, Canada, Australia, UK, New Zealand, Japan, China, and much of the EU (for permanent residency) almost always demand police certificates or character assessments.
• Rules change — What’s true today may shift. Always verify with the official embassy/consulate website or a local immigration lawyer for your specific nationality and situation.

Practical Advice. If you have a criminal record, countries like Mexico, Chile, Colombia, Cambodia, and France/Italy (specific visas) are among the more accessible for long-term residency without the upfront hurdle of obtaining an FBI check or equivalent.

Quite surprising really. Many of these are considered risky by the PPU, so not advisable to try it, while still on the SOR.