Up to your oxters (or my oxters, etc.) is a phrase I often heard growing up in County Mayo in Ireland. Oxter means ‘armpit’, normally, so up to your oxters means ‘up to your armpits’ – whether literally or figuratively. You could be up to your oxters in a river or in housework.
The word is used in dialects in Ireland, Scotland, England, and the Isle of Man. As well as signifying the armpit, it can refer to the underside of the upper arm more generally, to the fold of the arm when bent against the body, and to the armhole of a coat or jacket.
Oxter also has various verb senses. The OED lists these as: ‘to support by the arm, walk arm in arm with; to take or carry under the arm; to embrace, put one’s arm around’. It dates the earliest example to Robert Burns in 1796: ‘The Priest he was oxter’d, the Clerk he was carried.’ The noun is centuries older.
The etymology of oxter is surprisingly complicated, but the word is of clearly Germanic cast. From the OED: