Unfortunately, she asked me on a day when I had been spending time trying to get "very" taken out of a whole lot of places in the manuscript I was working on. So I suggested that taking "very" out of the sentence in favor of another word would be better. I offered her "I'm incredibly sorry," "I'm tremendously sorry," and "I'm terribly sorry."
But then, as I thought more about it, I started to think that maybe there are better options. For instance:
I'm unspeakably sorry.
I'm horrendously sorry.
And maybe some not-exactly-better options...
I'm serendipitously sorry. (For those times when you're sorry, just not about that.)
I'm herbaciously sorry. (For an apology with real pith.)
I'm supernaturally sorry. (As if there were a ghost of a chance about that.)
I'm ironically sorry. (Because you're feeling kind of flat, emotionally.)
I'm suspiciously sorry. (Since you're not really sure what went on.)
Well, you get the idea. It was enough to make me chuckle for a while on the drive home. But, then, it had been a long day and I may have been having blood sugar issues.
Even so, I strongly suggest that the next time you need to offer an apology, you find a good adverb toss into the mix. If the recipient doesnt find it funny, you can always go back to "very" in a pinch.