What actions will count against the 'I don't need no stinkin' upgrades!" achievement?

Solution 1:

It seems that the achievement trigger specifically looks at whether you have manually upgraded a system from the Ship's Systems screen, including subsystems that don't require extra power (Doors, Nav, or Sensors).

Random Events that give system or reactor upgrades (such as Federation Terraforming Team C12 or Slug and Rock Standoff) do not count against this achievement.

Likewise, purchasing additional weapons or systems (e.g. Drones or Hacking) will not count against the achievement. The Backup Battery is also perfectly fine to use (it doesn't provide permanent reactor power), and can actually be a good choice for trying to earn this achievement.