My experience was somewhat of an extreme case. I believed whole-heartedly in the company I worked for, had a buy-and-hold mentality, and believed in investing for the long run. I exercised my ISOs, paid the AMT, and then held them for a couple years. Eventually my company was bought out, by another company that had a lot of debt. Even though business was booming, we were hiring, etc., the decision makers saw an opportunity to strategically default - essentially reorganize in bankruptcy. Very abruptly, unexpectedly, and quite spectacularly, our stock price went from the sixties to the pennies, and then subsequently all the way to zero.
I was left with an almost unbelievable AMT credit. The stock I expected to sell and pay long term capital gains on no longer existed, so the AMT credit wasn't much use, except to offset my regular income taxes. It would have taken at least two decades, probably longer, for me to use that enormous AMT credit... but, an improbable, perhaps even miraculous bit of congressional lobbying made the "dot-com collapse" AMT credit refundable over a period of five years, and so uncle Sam paid us back in a reasonable amount of time. That said, I believe the refundability of the AMT credit was a one-time program. If you walk onto the AMT tight-rope today, I don't think there's a net to catch your fall.
Obviously mine was an extreme case, and is unlikely to happen to you - at least, not in that exact same way. That said, I would humbly give you some food for thought.
Not surprisingly, I believe that playing with AMT is akin to messing with fire. When channeled via a rocket, fire can get you into outer space! It can also burn your house down and destroy everything you own. I wouldn't take that risk again.
However, I unequivocally believe that diversifying through the controlled divestiture of ISOs/company stock is very very important. I was literally a teenager when I was given my ISOs. A very wise (and much much older) product manager that I worked with tried, tried, and tried to sell me on the notion that having so many eggs in one basket was a bad idea. I didn't listen. I said "this is the new economy, all the old rules are dead, this time it's different", etc... He ended up with an amazing condo with a view of the SF bay, and I kept living in rent controlled apartment in SJ.
Much like you would dollar cost average into the stock market, I would always be exercising/immediately selling a block of ISOs, every 6 months, paying short-term capital gains taxes and reinvesting what's left into your broadly diversified index fund portfolio. If you plan to stay with your company over the longer term, you can exercise/sell your ISOs in a similar manner as your original vesting schedule. For me, I vested 25% every 4 years, so if I could do it again, I'd exercise and sell 25% of all my vested ISOs each year. Really it could be anything you're comfortable with. (Note that if you intend to leave the company in the near future, the "what do I do with the ISOs" becomes a very complex issue.)
You probably live in a HCOL state and pay lots of taxes, and it sucks handing over such a huge chunk of cash to the government. But... look at it this way. If you exercise and sell immediately, you pay short term capital gains, obviously.... but, you know the dollar amount difference between the STCG and the LTCG rate? The money you lose using that strategy? Well, that dollar amount actually buys you: 1) insurance against the evil destruction that could result from the AMT, and 2) the ability to immediately diversify into another investment that doesn't carry the same concentration risk as your company stock holdings. If I could go back and do everything again, I would pay that extra cost with a smile on my face every single day.
All that said - these are my opinions, based on my sole experience. I would encourage you to listen and consider all the advice your given, especially from anyone who says I'm full of crap.
One last thing. If you ever do decide to exercise, pay AMT and hold for 365+ days.... I recommend you immediately sell (right after the exercise) enough stock to pay for: the cost of exercising the options, the AMT bill on what you'll hold, and short term capital gains taxes on the stock you're selling (to pay for all 3 of these). If you do this, you make the stock you exercise pay for itself, at the very least. If you don't, there's a scenario where you use your cash/savings to exercise the options, sell nothing, and then watch the value of the stock drop below the total dollar amount of AMT you'll owe. Even if the stock price rises again post-AMT, digging into your cash/savings to pay taxes on an asset that's worth less than the tax bill would really suck. Trust me on that one.
Oh, and congrats on your IPO!