Learn Russian: Наша компания не допускает даже наименьших ошибок. - Our company does not allow even the smallest mistakes.

It uses the prefix наи-, which creates an absolute superlative meaning "the absolute most" or "the very". Here, it is added to меньших, the genitive plural of меньший (smaller/smallest). Together, наименьший means "the very smallest".

Can I attach the prefix наи- to any adjective to mean "the absolute most"?

No, you cannot add it to standard adjectives like красивый. The prefix наи- is almost exclusively added to special comparative or superlative forms, most commonly irregular ones like лучший (наилучший), худший (наихудший), больший (наибольший), and меньший (наименьший).

Why is наименьших ошибок in the genitive case rather than the accusative?

This is due to the genitive of negation. When a transitive verb is negated with не, its direct object often takes the genitive case instead of the accusative. This is strongly preferred for abstract concepts like ошибка and verbs of allowance or possession like допускать.

Why use допускает here instead of разрешает for "allow"?