As much as possible; to the greatest extent that is feasible (often used in written or formal contexts).
出来うる限り早くご連絡します。
I will contact you as soon as possible.
出来うる限りの準備は整えたつもりだ。
I believe I have made all the preparations possible.
出来うる限り参加したいが、仕事次第だ。
I want to participate as much as possible, but it depends on work.