飽(あ)くまでも

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', and も 'even', 飽(あ)くまでも will imply that (A) is being done 'persistently', 'stubbornly', or 'to the bitter end'. The more literal translation of 'to continue (A) even to the point of getting tired of it' fits these interpretations quite well.
飽(あ)くまでも will often come at the beginning of sentences before going on to describe that thing that is being done persistently.

• 容疑者(ようぎしゃ)はあくまでも自分(じぶん)がしたことを認(みと)めようとしませんでした。
  The suspect stubbornly refused to admit what he had done.
• 私(わたし)たちはあくまでもあなたのお子(こ)さんを見(み)つけるつもりです！
  We intend to look for your child until we can find them!
• 彼女(かのじょ)はあくまでも子供(こども)たちには秘密(ひみつ)にしておこうと思(おも)っていたが、これ以上(いじょう)子供(こども)たちには嘘(うそ)をついていられないと思(おも)い、子供(こども)たちに全(すべ)てを話(はな)した。
  She thought that she would have kept it a secret from her kids till the bitter end, but she couldn't lie to her children any longer, so she told them everything.

も is not required, and 飽(あ)くまで may sometimes be seen without it. However, as the phrase is quite strong, it is far more common for も to be included to highlight the emphasis.

• 犯人(はんにん)のことなら私(わたし)にお任(まか)せください！あくまで彼(かれ)を捕(つか)まえるまで探(さが)し続(つづ)けるつもりです！
  Leave it up to me to find the culprit! I intend to look for them until I catch them!

Due to the persistence that the verb 飽(あ)く itself can highlight, it will often be seen together in sentences that include words like つもり 'intention', to show that someone intends to do something regardless of the consequences.
Despite 飽(あ)くまで being a combination of 飽(あ)く and まで, it will appear in most dictionaries as a standalone adverb in modern Japanese.