Disclaimer: I've reached and fought through most of tier 2, but I haven't done any of the endgame (tier 3) yet. However, I doubt anything changes significantly.
The short version: Yes, but it will take you a lot longer than if you spent some money for energy.
The long version: Everything in this game is available via normal FTP gameplay. However, there's basically a limit of how much you can play (about an hour per day) due to the 100 mist energy limit. This is enough to get through a tier, with some leftover energy to revive you should you die. However, what isn't immediately obvious is that the energy needs to revive on death increases per death (in one trip, it resets for each trip), AND for deeper tier. So, that extra 20 energy you could get 4 revives out of in tier 1? (IIRC, the energy needed per death is 2,4,6, and then 8) To get 4 revives in tier 2, you'd need a whopping 75 energy, which would severely limit how many levels you could explore. (The amount per death is 5,10,20, and 40.)
Additionally, there's the problem of crafting. Everything up to 2 star equipment (what you need for tier 2) can be done with just mist energy, as the 2 star crafting recipe requires 50 energy. However, the 3 star recipe requires 200 energy, and the later ones are worse. (A 4 star craft needs 400 energy, and I think 5 star is 800.) Thankfully, you can buy crystal energy with ingame money, but it's not cheap. When I first joined (shortly after the game came to steam), the price was around 3500 crowns per 100 energy. As of this writing it's risen to 4800 crowns, and since I read that it was over 7k crowns before the steam update, I have a feeling it's not dropping any time soon. I'm finding that getting the energy for crafting is as much of a problem as getting the materials and the recipe. (For the 4 star sword I'll be crafting soon: 10,000 crowns for the recipe, 2,500 crowns required in the craft, I found the materials in the caves, and the 300 crystal energy I needed cost me another 12,000 crowns. That's not even counting the full day's worth of mist energy I'll need.)
So to recap, buying energy won't get you anything you couldn't obtain via normal gameplay, but it'll really cut down on the amount of time and effort you'd otherwise need to craft stuff. (One last note: You can also sell crystal energy ingame for crowns, but I'd personally rather obtain those via normal gameplay.)