Yes, there will be a time when this strategy does not work as well. Typically, big companies have salary bands per level which are set by HR (based on market analysis). While managers have some discretion on top (and may have more when it comes to RSUs), that discretion is somewhat limited--especially for those just starting out in their careers. Hence my comment that eventually it's just better to get promoted.
Does it leave a bad taste in your manager's mouth? I don't know. That depends on your manager. ;)
On the one hand, there's absolutely nothing wrong with demanding a fair price for your labor. You only have one life, and you're choosing to give a big chunk of it to your employer! And if you're underpaid, competing offers are great evidence to rectify that!
On the other hand, don't make a threat you can't back up. If you told me your plan was to get competing offers that you actually find attractive and then take them if your present employer can't match them, I'd say that's a great plan.
If you like your job and your team and don't want to switch, however, then I think presenting competing offers on a yearly frequency presents two potential challenges:
1. Your manager at some point may say, "Can't help you." They're not gonna fire you, but they may realize you're bluffing, so you won't be able to use this tactic in the future!
2. It way detract from talking about other, ultimately more important facets of workplace satisfaction, like working on projects you like, working with teams you like, etc.