the short answer is: No. you do not HAVE to pay $125,000.00 at the end of your first year. that is only the amount IF you decide to exercise.
*fine print: But if you leave or get let go (which happens quite frequently at top tier Silicon Valley firms), you lose anything that you don't exercise. you're basically chained by a pair of golden handcuffs. in other words, you're stuck with the company until a liquidation event such as IPO or secondary market selling (you can expect to spend a few years before getting anything out of your stocks)
Now, it's hard to say whether or not to exercise at that time, especially given we don't know the details of the company. you only should exercise if you foresee your quitting, anticipate getting fired, AND you strongly feel that stock price will keep going up. if you're in SF bay, i believe you have 10 years until your options expire (at which point they are gone forever, but that's 10 years and usually companies IPO well within 7 years).
i would recommend you get a very good tax advisor (someone that understands AMT and stock options tax loopholes/rules like the back of their hand).
I'm going to take a long shot and assume that you got an amazing offer and that you got a massive amount of ISOs from them. so i'll give this as an advice - first, congrats on owning a lot on paper today if you're still there. you chose to be an early employee at a good tech company. However, you should be more worried about AMT (alternative min tax). you will get enslaved by the IRS if you exercise your shares and can't pay the AMT. suppose, in your fictional scenario, your stock options increase 2x, on paper. you now own $1 Mil in options. but you would be paying $280000 in taxes if you chose to exercise them right now. Now, unless you can sell that IMMEDIATELY on the secondary market, i would highly advise you not to exercise right now. only exercise your ISOs when you can turn around and sell them (either waiting for IPO, or if company offers secondary market approved trading).