Almost; very closely; very nearly.
Whoops, Bruno pulled another paper towel. The whole procedure was repeated. Well-nigh a third of the vegetable marrow mass was now gone! (Context: A boy named Bruno hates the meal served for lunch, so he wraps pieces of it in serviettes and throws them under the table. After this latest repetition, almost a third of the food was successfully discarded. Source: J∙ Okram - The Mystery of the Rammed Key.)