Thrice is the limit of how often you want to call HR in a week, unrequested. In a busy shop, this person is likely not able to move your particular case along all that quickly.
One technique is to say "when do you think you will have the next step ready?" - in this case, the approval. The HR person should be able to say "it usually takes X days" and/or "you case is unusual because..." (for example, I had a case where the executive who needed to sign off was trapped in a crazy long off site meeting and couldn't be found for 3 days). Don't call back until AFTER the date the representative has given to you.
After several days of 'any day now' - start questioning the veracity of this statement. "Hey, we've been talking every day for the last X days... I'm starting to disbelieve that it's likely to change".
There also comes a point - after about 2 weeks of no progress - where I tend to give up. This is largely variable based on the market for your skills - if this is the one job offer you're likely to get - keep chasing. But if you have marketable skills, figure that this company isn't sufficiently mobilized to do hiring and find a position that will actually close the deal.
Also - no offer is real until it's fully approved and handed to you in writing. Don't stop looking, don't figure that your search is over - but as much effort into interviewing as you do in calling the HR rep. Worst case scenario on that is you may have TWO job offers - that's a great position for YOU!