There are two sources of energy in the game: mist and crystal. Mist energy is 'free' and is capped at 100. Crystal energy can be purchased from other players with in-game crowns or from the developer with US $. Various actions in the game that cost energy consume Mist energy, if available, before consuming Crystal energy.
Can I play the game without ever purchasing Crystal energy and still experience nearly everything someone with unlimited Crystal energy would?
Since you can purchase crystal energy with the in-game currency, it seems like kind of a moot point. There are times where you will need more than 100 energy, for crafting or such. Now you could choose to just not do this, and then you wouldn't need crystal energy. But you can also choose to go farm some crowns, then buy the crystal energy that you need.
So you can certainly get through the game without ever spending real money, and if you aren't spending real money, you should just look at crystal energy as another in-game resource which you occasionally need to replenish with crowns.
(disclaimer, this answer is based on 4 days (400 mist energy) of game play)
You can theoretically experience the entire game without paying any real world money. However, with 100 mist energy per day, you are stuck running only one full dungeon per day (~8 levels). This will not leave you with enough energy to craft anything of note; crafting takes 50 energy for 2star items for example. This means that one day you run dungeons, the next day you craft a couple of items. Additionally, each death and respawn in a dungeon costs energy at an exponentially increasing rate; 2 for the first respawn, 4 for the next, then 8 and so on. With the 100 energy maximum, you're going to be in a tight spot when reaching higher tiers of dungeon which can be rather challenging from what I have heard.
However, you can buy energy using in-game currency. It currently goes for about 3700 credits for 100 energy. After running 300 energy worth of dungeons, I racked up about 11,000 credits and haven't even begun to use the auction house to sell my approximately 30 assorted crafting materials (which are worth 15,000+ in total). Now I've had to use most of my cash to buy crafting recipes, but in theory I could be playing the auction house to get the credits needed to get another 400 or so mist energy, run more dungeons, and so forth.
Short answer, yes you can experience the entire game with the free energy, but it will be slow.
A lot has changed in Spiral Knights since this question was asked, so I'd like to offer a more modern perspective. Energy (and promotional packs featuring Energy) is not the only product available for purchase anymore, however it still is the primary currency for paid content. In addition to Energy, paid context also exists in the form of the items listed below.
All of these changes and we still haven't mentioned the July 30, 2013 patch which has effectively changed the pay model of Spiral Knights by removing energy fees from elevators and crafting.
So with all of that groundwork out of the way, is it possible to experience all content in Spiral Knights without purchasing crystal energy? Yes.
The July 30 changes fundamentally change the role of Energy from that of a content budgeting mechanic (or in other words, restricting how much content you could experience at any given time) to now primarily existing to expedite content to players with more money than time. That said, these changes have not necessarily made progression faster. With respect to Tier 3 progression, I have experienced a slowdown in leveling items simply due to the gambling mechanic in the Forge and the drop rate of rarities. So what in essense has happened is that players will now be able to put in more hours played, and may end up progressing faster than they would have in real time by only being able to play so many levels per day. None of this has restricted content to free players.
We're not quite done yet, though. What about all of that other paid content I mentioned earlier? Is it possible to experience all content in Spiral Knights without purchasing ANY content? Sort of.
To answer that thoroughly, we'll need to take a glance at Spiral Knight's integration with Steam. Spiral Knights features quite a lot of Steam integration -- the game is among the few participants in Inventory Trading, was among the first F2P titles on the platform, and features Operation Crimson Hammer and the Guardian Armor Pack as purchasable content in the Steam Store. The interesting thing about items that you can purchase in the Steam Store is that they can be kept in your Steam Inventory and traded with other players. F2P users can't purchase this content in Spiral Knights with CE, so Steam trading is the only way to acquire these for free.
The caveat to this is that there are some restrictions to steam trading that the player should be familiar with, so that they aren't stuck waiting two weeks for Steam Guard to clear them.
Update: As of September 25, 2013, the Operation Crimson Hammer DLC can be purchased for 3200 Energy from the Missions interface.
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.)
