I have started playing Minecraft quite late, which is to say, I started two days ago with v1.0, but I am slowly starting to discover the addictive appeal.
However, from looking around and talking to other players it seems the majority are most interested in big building projects, intricate constructions, processes, etc. I have to say that this all actually interests me very little, since even if I did have the patience for such big and complicated undertakings, I would lose interest much too quickly since it's still pretty static in the end. I love to explore big worlds others created, but wouldn't want to build one myself.
What I enjoy most are really just the very basic aspects Survival and Mining. Survival is probably the reason why I often feel compelled to start a new world. The very first few days of exploring a freshly generated world, gathering resources, finding a good, safe, practical location for a first modest home and building on it. That's an aspect that I enjoy in a very general sense, but I can think of no other game that has done such a good job at it (if you know any, please recommend them!)
The Mining aspect is pretty easy to rationalise. Exploring caverns, finding treasure, defeating monsters; the basic ingredients of every traditional RPG. I don't know how long it will take until I grow tired of it, but digging and digging until suddenly reaching a huge cavern with a waterfall dropping from the darkness above is just great. Even more so since I know that it's all more or less randomly generated, every world is unique, and the generation algorithms will probably be extended and improved as time goes on. I'm much for game worlds designed by talented, creative people, but if they are as beautiful as Minecraft's there's a lot to be said for completely unique, automatically generated environments.
In terms of building, I stick to the basics mostly. A nice home with a few practical rooms and a garden. Deep tunnels and shafts for mining. Bridges, cattle pens, gardens--and strategically placed emergency exits in case a creeper is lurking on the front porch! I learned that just today after one blew up my little garden and half of my house's front. The beautiful worlds are one thing, but creating little structures that adapt to the scenery is great. I'm thinking for example of the little railed bridge, beautifully illuminated at night, that leads me from my house to my little cattle ranch. That's probably why I don't really want to bother with huge mega-structures. I have much more fun building on what's already there.
Within its own extremely low world resolution and the very few building blocks, I find few other games if any have ever given me this amount of liberty in making my dwellings look the way I want them to, which I think is a basic impulse that many games could profit from. I always liked the complete freedom in designing my villages in the Clonk games (which Minecraft reminds me heavily of), but although I wouldn't have Clonk any other way, its 2D nature puts certain limitations on it, as does the fact that the structures themselves are pre-built.
So, how do you most like to play the game, and what motivations are behind your way of playing?
I played the game quite intensively about a year ago, and for about a week last summer. I too loved the exploration part. You can read about my humble beginnings in the review I wrote (one of the few times I felt compelled to write a review).
Most time I spent exploring one really humongous cave-system. I got lost so often that I started making signs and placing them at every junction, numbering them and signaling which way the sections went.
Last summer I spent more time building (together with some friends on a private server). Building is fun also, but if you want to take on a big project you need a lot of resources and that requires systemic mining, which can become boring at time.
It's amazing in how many different ways the game can be played. I see a lot of different people enjoying the game in a lot of different ways.
The way I played was fairly organic. I started with a mine and dug to the lowest point of the map. Then, I built a tower that reached to the highest point. Then I built an extravagant stairwell that climbed to the tower and a fountain at its base. I sculpted the mountain the tower was on to be an extension of it. I then built a tremendous wall around the tower and left room for what I planned to be a small city. I built a port for my city and a lighthouse for when I was boating around. I built a rail system up to what I planned would be the site of another city. I looked at my town and decided that the land around it needed to be cultivated, so I built some farms.
It just goes on and on like this. Basically I look at something and see that it would be a great place to build a thing. Then I build it, but usually get distracted by another something to build a thing on. Eventually I wind up with my own little world.
I haven't played extensively since the last few updates came out, but I like what I see. I'll likely pick it up again when I'm willing to forfeit of month of my life.