いかなる償いもいたすつもりでございます。
Ikanaru tsugunai mo itasu tsumori de gozaimasu.
I’m prepared to do whatever is required to make amends.

This short sentence represents an extreme declaration of formal apology. The Japanese language is said by some to reflect a culture of apology, in the sense that the social act most fundamentally enshrined in the language seems to be the apology. Even the most ordinary transaction—asking for information or making a minor request—usually incorporates such standard phrases as sumimasen (“I’m sorry”) or osoreirimasu (“I beg your pardon”), expressions that, at least on a literal level, convey apology. You might say that the impact of such apologetic expressions has become diluted through overuse. As a result, when the primary purpose actually is to formally and earnestly apologize for something one has done, the language employed tends to be extreme. What’s more, people may literally get down on their knees or even prostrate themselves to demonstrate the sincerity of their words.

If you really want to reach back and humble yourself the way they do in samurai movies, you might try this (though you’ll probably have to explain it after having used it):

七重の膝を八重に折りまして、お詫びいたします。
Nanae no hiza o yae ni orimashite, owabi itashimasu.
I beg your forgiveness on bended knee.

A literal translation of this standard expression will provide an idea of the extreme attitude it expresses. Nanae no hiza o yae ni orimashite means forcing one’s knees (which bend only in two) to bend seven ways (nanae) and then once more (yae, “eightfold”)—as if this were physiologically possible—from which decidedly uncomfortable position one offers one’s apology (owabi itashimasu).

Japanese society has long honored a sort of tacit understanding whereby a transgressor can effectively expiate his offense by making a sincere and sufficiently humble apology. In addition, a direct apology is considered the most reliable way to assuage the anger of the offended party. While both these functions have helped established the spoken apology as an essential social lubricant, it no longer necessarily holds the power it once did. These days, even the most sincere apology may be answered by a demand for complete financial compensation.

In the unfortunate event that you have caused an injury through your own negligence or carelessly started a fire that spread and damaged other people’s property, no apology will be sufficient to undo the harm. Nevertheless, by conscientiously making the most humble kind of apology, you can at least demonstrate your sincere regret, and that is perhaps the first point to address. Should you commit the sort of offense that affects another person’s reputation—calling off the wedding after the invitations have been sent out, for instance—be prepared to grovel your way through an apology that will surely be a once-in-a-lifetime lesson in public humility.

– Source: A handbook of common Japanese phrases –