Really depends on the type of story.
I can cheat and answer "game", because games can be any of the other medium as well (except for theatrical play)

I'd most likely be writing novel or theatre. I could also imagine myself writing radio drama (drama CD). Nonetheless I voted for live-action TV. Had I the skills, money and staff, I'd love to do a TV miniseries.

Hmm, it would be something so friggin drama and tear-jerking. Something like 5 Centimeters per Second (Anime), in a form of musical would be awesome I guess.

A novel.
But i'd love if it could become an animated movie!

A romcom movie... wait I've done that already...so...

Novel. Full fledged fantasy novel. Something in the scope of Ice and Fire saga. My own magnum opus.

Where's the Visual Novel option?
And don't say that it falls under the "Game" category, because it doesn't.

I have several ideas suitable for many mediums.

I was going to say manga, but...

right now, I'm in disney phase, and recently I also saw Lotr animated, and I thought, wow, that's horrible. So I was beginning to imagine how I would do that, animated. So yeah. Also, an original animated movie would be great. but 2d, not 3d. not because of skills.

Definitely a game. With unlimited budget, a game would create an interactive experience like none other. A plot with infinite possibilities and choices would be amazing.

Definitely game. I'm a gamer through and through and as such all of my ideas tend to be geared towards the gaming medium. It would be awesome to be able to create any game I could think of without restrictions.

I went for animated movie, mainly because it's the field I sort of work in, so it makes sense in my head.

Hm....this one is really hard considering my enjoyment of multiple forms of media......

I really had to think hard about this one. But if I really could make the "perfect" whatever, I'd choose a TV series. It would be extremely well plot out (no filler moments), great actors, and have a definite end. Every moment would be guaranteed to be intense!

Though, realistically speaking with my abilities and circumstances, I'd probably only be able to write a novel or a mildly crappy art manga.

It all depends on the story. Sometimes it works better as a movie, game or whatever. There is no such thing as an absolute medium when it comes to entertainment, you have to adapt to time and audience.
Let's say Tekken was supposed to be a book at first. It could be done if the author's skilled, but its current game form is obviously the optimal.

Well, a novel first. So I have everything planned out (I'm in the midst of doing that for one of my many projects, called "The Engraved".)
After that's finished I plan on drawing a comic version of it (I already did that with the prelude of the novel, but I plan on redrawing it >.>...)
And while I'd like to see it animated or live-action, I know it's probably never going to happen so I won't get my hopes up XD.

WTF is a drama cd? If it's just some crap where people are recorded trying to act, we kinda of got out of that around the 1950's radio shows.

If I had an original story to write, I'd use the medium of a book. Comics are for kids, same goes for manga. IDGAF what anyone says, but it's pretty much the trust. At the very least, they're made for immature minds that need pictures because too much text bores them (that does include me, but I do read more books than manga, by far). If I were to write a story, pictures couldn't depict the twisted shit that flows forth from this mind if mine. If they somehow could, most stores wouldn't sell it.

I would love to write a song someday. (:

where would a music video fall under? i dunno, that was my first thought and i voted for live-action movie, even though in my intended future career i'd love to write novels XD

A novel.
Which it would be adaptated into manga, anime, live action, drama cd, theatrical play etc.

What about drawings/pictures/images/art...

I would make a game. A free roaming open world RPG to be more exact. To be even more exact. A life simulator where you take control of your character you made and live his or her life.
The setting:
It's set in a dark high fantasy world during late summer of year 51 of the 15th era or year 1551. It is the beginning of the Raven war in the Empire. The Raven war is a internal conflict of a clan on who to take the seat of clan head as the previous one died without a strong heir. However, this conflict pulled the entire Empire and all its clans into the conflict dividing it into two. 12 major clans and 43 minor clans battling it out. A total of 63 million troops. You say total war? I say yes, indeed.
You would start in a tavern. You'd wake up as you hear screams of terror filling the village and fire raging through the tavern. As you get out of the tavern you find the sky lit up by fire arrows and larger projectiles flung everywhere. You'll see two massive armies (approximately 500.000 each) and the village is in the middle of the battle. No one cares about civilian casualties.
Well, anyway, as you get away the goal is up to the player. If you want to stop the war you can do that. Take advantage of it by plunder and/or exploitation also works. Live a simple farmer's life? Can do. Merchant? Blacksmith? Hunter? Beggar? Thief? Mercenary? Tavern owner? Housewife/man? Wizard? Prostitute? Pimp? It is all okay.
The war will always have the same outcome unless you join in on it yourself which you then can have some influence on the outcome. By outcome i mean that the same side will always win. Anything in between can happen differently. In your first playthrough one battle may happen at the beginning of year 53 and in the next it might be the end of the same year instead or maybe it never actually happen or even 10 years later.
Your character will of course age by the year. So when your character turn 50 he/she will look somewhat like 50.

As for how the game works. Stairs, hills, slopes etc all have an animation instead of showing the same running or walking animation.
The amount of weapons, cloths and armors will be quite massive (50-ish weapons one-handed two-handed and ranged. 6-ish types for each armor tier with 10-ish designs and tons of shields and cloths) with each weapon having their own animations and each armor tier (cloths, light, medium, heavy) having their own animations as well (heavy would be hard to run in and women wouldn't be able to walk like on a catwalk in them) while light armor would be more agile but not as much movement as clothing or being in the nude. Medium would be in between light and heavy.
There'd be heavy character creation as well as a randomize button and presets. The randomize button would of course be within reason and not give you a character without a chin, a 2 feet long nose, alien eyes, mouth without lips and wearing boxers with heart patterns, diving fins, a clown nose and chocking pink liberty spikes for hair.

as for the leveling. Elder Scrolls style I guess with each level giving you points to put into Strength, Dexterity, Constitution, Intelligence, Wisdom, Charisma and Perception.
What I mean by Elder Scrolls style is that using a sword will make you better with it as you use it just as running will make you better at running and jumping will make you better at jumping etc etc...
Perks will be unlocked as you level your skills while putting points in attributes unlock what you can use. Don't want to see a scholar with strength of 5 to wear heavy full plate armor flinging a giant halberd around like it's a toothpick.

Another thing. When your character dies by old age, illness or some other way while having kids you can choose to play as one of his/her children from a certain age. Your choice on how that character live is once again up to the player.

This just became quite long. Better cut it here then.

That's an interesting idea and I've heard similar before, but even if you had limitless money, staff and skill there would be other restrictions. What kind of engine do you think would be required to run that? Also those troop numbers in a high fantasy world are just insane.

I like that generation idea, though it would need a lot of work to make it practical. Another negative thing is that a games without a coherent story (which your game would inevitably become) have been proven not to be interesting. It does have potential if you break out of the standards. There have been many of those of course, but not any that I can think of that still have such a standard setting.

The troop number is just the amount of currently recruited people across all the clans and they wouldn't all be in the same place. It's no "save the world from evil" scenario now, so the most active clans would obviously be the Raven clan (FYI they're called the Raven clan as the Raven is their symbol) and their closest allies. The rest would be quite casual about it unless called into battle by the war leader of their side (which would be either of the successors of the Raven clan).

As for the engine. It would be built from scratch to be optimized for such a game. Let say that most civilians would be quite generic and boring like Assassin's Creed and so would most of the armies. And when you engage in battle with them only those you fight with will be enabled for heavier AI code and stuff. And the maximum amount of drawn NPCs would be something like 1000 if generic and perhaps 50-ish non-generic. This of course would depend on the machine it runs on. The generic part would also depend on the NPC. Guards, merchants and other needed NPCs would be different.

I can understand that it can be seen as a incoherent story or even without on as it mostly is, but the actual story is the war told from one person's perspective and that person's life and how it changed for better or worse during that time. Think of it as a machinima game or along those lines. The player tell the story of their character in my world that is constantly in a chaotic state. There would be tools like a history writer which automatically writes down what happened each day (PC plowed a field from x minute to y minute from coordinate xyz to coordinate xyz while state of war is x). There'd also be some backtracking video player which play these events from a history file so people could record and make videos or even take screenshots to put up with the story.
My story would just be the lore and the standard Raven war with its events. Where your character is or isn't may effect my story but also yours. You may once again be squeezed in between two faction of a battle or even a months or years long siege. Maybe you'll even be kidnapped by bandits and sold to slavery.
Think of it as a *your input may or may not change the course of my story but my story will always change yours* type of thing. An extremely interactive story so to speak.