It's basically just a number.
You can have it in many formats.

• Binary
• Decimal
• Hexadecimal
• Wif
• Mnemonic

So it depends on what you want format you want to remember or save.
But since you said:

When we start with the bare minimum.
The name BITcoin tells you something, a bit is either one or zero, true or false.
And the Binary key is 256 Bits, so that means it's a key made up of 256 one's/zero's.
If you were to flip a coin 256 times and write down heads as one and tails as zero you will end up with a valid Binary key.
You could consider that a paper wallet already but it is possible to derive more then one address from this 256Bit key so you can say it's kinda incomplete and there are better key formats available, this is just the bare minimum you can use to be able to access your funds.

In Binary form it's called BASE2 because it only contains ones and zeroes.
Here is an example:

But you can also convert this key to for example BASE 10, which is the decimal system that we use.
Then it looks like this:
So you can also write down that number if you wanted to save your private key.
In essence, this number protects your funds, it is just too large a number for anyone or even a computer to guess. 

You can also use a more preferred base which is BASE16.
BASE16 has '1234567890ABCDEF' in stead of 1 and 0  or 1234567890 and this makes the key a lot shorter.
In BASE16 the key looks like this:

Either one of these 3 key types are enough to be able to access your funds but Bitcoin itself uses more specific key's that have additional compression information as well as checksum to prevent you from making a mistake these private key's are called WIF or Wallet Input Format and these can be directly imported into your wallet software.

Also, because you can make more then one address from one raw key, there are several types of WIF key's.
The raw key's above translate to these WIF Keys:

The one with the '5' prefix is referred to as the uncompressed WIF key, while the other one starts with 'L' or 'K' and its is referred to as the compressed WIF key.
These are the keys you will find on a printed paper wallet, usually in the form of a QR Code.
You can scan them with a bar code reader and the wallet application will usually pick it up from there.
 
Additionally you can turn the number into a Mnemonic sentence which turns it into words which will be easier to remember and write down.

I don't know if you also want to manually derive the addresses and draw QR-codes but that doesn't fit the term minimalistic anymore