I wasn't paying too close of attention, just opening the game every 20-30 minutes to buy things and closing the game again. I think I was able to buy The Big Cheese before the last few in the screenshot, allowing me to activate Challenge mode without buying everything.
I didn't want to get flagged as a cheater so I waited until I had enough to buy everything before activating the challenge mode so I'm not sure what would have happened if I tried to do it early.