飽（あ）く is the old-fashioned and somewhat poetic version of 飽（あ）きる the る-Verb meaning 'to tire of (A)', or 'to get sick of (A)'. When used in conjunction with the adverbial particles まで 'until', 飽（あ）くまで can imply that (A) is 'just', 'only', or 'strictly' something. The more literal translation of 'to continue (A) to the point of getting tired of it' fits these interpretations quite well, implying that (A) will never deviate from what is being described about it. This use of 飽（あ）くまで often appears at the beginning of a phrase, and functions adverbially. The adverbial particle も 'even' may be added for extra emphasis, but it is not required.

• 彼（かれ）はあくまでも新人（しんじん）なので優（やさ）しく接（せっ）してあげてください。
  He is merely a newbie, so please treat him kindly.
• それはあくまで可能性（かのうせい）に過（す）ぎないからなんの役（やく）にも立（た）たない。
  That is merely a possibility, so it doesn't help.
• 私（わたし）はあくまでも素人（しんじん）に過（す）ぎませんが、設計（せっけい）にミスがあると思（おも）います。
  I am just an amateur, but I think that there is a mistake in the design.

In many cases, 飽（あ）くまで will be used in polite sentences when the speaker is expressing their own opinion. In these cases, the nuance of 'this is merely my opinion' is conveying to the listener that the speaker realizes that what they are saying is not the only viewpoint.

• これはあくまで私（わたし）の意見（いけん）なので参考（さんこう）にしないでくださいね。
  This is just my opinion, so please don't refer to it.

This way of highlighting opinions is often combined with structures like けど, or が when the speaker wants to soften their statement before offering what they think a solution might be.