Lets say I have a currently running screen session I am interacting with through putty. I\'ve realized that the scrollback buffer is too small and would like to increase it
For posterity, this answer is incorrect as noted by Steven Lu. Leaving original text however.
Original answer:
To those arriving via web search (several years later)...
When using screen, your scrollback buffer is a combination of both the screen scrollback buffer as the two previous answers have noted, as well as your putty scrollback buffer.
Be sure that you are increasing BOTH the putty scrollback buffer as well as the screen scrollback buffer, else your putty window itself won't let you scroll back to see your screen's scrollback history (overcome by scrolling within screen with ctrl+a->ctrl+u)
You can change your putty scrollback limit under the "Window" category in the settings. Exiting and reopening a putty session to your screen won't close your screen (assuming you just close the putty window and don't type exit), as the OP asked for.
Hope that helps identify why increasing the screen's scrollback buffer doesn't solve someone's problem.