Why do we need base money at all? For example, we could imagine an arrangement like this:


There are currently \$0 in the economy.
A business takes out a \$50 loan. The bank sets there bank account to \$50 - \$50 debt = \$0 net.
The business now pays it workers \$40 to make stuff (through the bank).
The workers in turn pay back \$45 to the business to buy stuff (through the bank).


(We can imagine there being a lot more steps)

Etc...
We can use bank notes instead of currency


What is the purpose of base money then?
TL;DR You have defined a de facto currency with that bank as a central bank.  It is generally considered better to have everyone using the same currency. 

In theory, what you suggest can indeed work, and has been used for period of time in various places.  Legally speaking the Bank of England doesn't mandate a minimum reserve, and so in theory a bank could operate with zero reserve (The EU mandates 2%, the US 10%).  In practice each bank must agree their own reserve ratio with the BoE, and the bank would likely laugh if someone said "zero".

You system in essence works, and all you have done is created a currency tied to that bank (lets call it Bank Z).  The problem is that no-one can actually pay anyone who isn't a customer of Bank Z (for more on electronic transfers see: https://economics.stackexchange.com/a/6170/119).  Paying someone who banks with Bank Z is easy, just move the money from your account to theirs, but if they don't have an account we can't give them any money - because we don't have any.

What we could do is give them an IOU - "if you open an account with us, we will credit it with Z\$X".  This would be a bank note for Bank Z.  Of course the value in that note is tied to our faith in that bank.

Suppose now we imagine Banks X, Y and Z operating in our economy.  What happens when Xavier (who banks at Bank X) want to pay Yvette (who banks at bank Y)?  Well neither bank can change the customer account of the other, so both customers are given IOUs (bank notes) and take them to the other bank.  Should the other bank accept these notes and increase they account accordingly?   Well that depends...

If both banks are both well used then sure, why not.  But suppose instead that 98% of people bank with Bank X, and 1% with bank Y - whose IOU would you rather have?  It is much easier to spend Bank X money, than Bank Y money, so that in itself could cause bank X money to command a premium.

If X\$100 = Y\$101 then clearly I will charge you more Y\$ for fixing your roof that X\$.  This can quickly spiral and before you know if Y\$ are worthless - they have suffered hyper inflation and been inflated away.

You have effectively created a state with multiple currencies, and in such situations, one currency would be expected to fall into disuse and become worthless.  Historically countries have often actively defended their currency from the use of external currencies by imposing limits on how much currency can be brought in.  It can be a real problem if your people are trading using dollars instead of your national currency, for the same reason.  If dollars are more useful, then why use local currency, and so it decreases in value.

Your set up is particularly fragile, because the banks are merely lightly regulated corporation, not the government.  The government can mandate that all government workers are paid in local currency - in effect it can force itself to have a large number of customers.  It is not a coincidence that the central banks are the bankers to the treasury of their respective countries.

As a government this is also a pain - you have multiple currencies, and you don't control any of them.  Your economy is not in your control, which has cause problems in the past.

Gold backed?  Assets?

I deliberately haven't talked about this part, as you don't really address it, but I suspect many will think first about it.  The backing and assets of the bank are really secondary to this discussion.  Their is nothing to stop all our banks saying that their dollars are pegged to the value of gold.  This simply means that you can, on demand, exchange your IOU for gold.  So long as nobody does this then you are fine.  Equally, even if a few do, you can buy they gold with your own IOUs and so that is still fine.

Any assets your bank has, can be used to "back" your currency and add strength to it, even though you have no "reserve" with any central bank.  This will help you.  Bank Y could respond by advertising that it has tons of gold in the vault and owns lots of houses and farmland, so it will be more likely to be able to honour it's IOUs that Bank X can.  This could turn the tide back in your favour, as people lose faith in X\$ and switch to Y\$.

Remember, even if your Bank Z starts with zero assets, it will make a profit in Z\$ over time, which can be used to buy things and so it will accumulate assets with time.  None of these need become liquid reserves.

Examples?

Your case is a bit too extreme, but near examples are worth a look

The biggest example that springs to mind is the Free Banking Era (aka "Wildcat Banking") in the US from about 1837-1862.  Andrew Jackson had "defeated" the Second US central bank and believed that banks could just be allowed to run things on their own.  The average life of a bank in that era was 5 years, and each bank's currency was worth a different amount, depending on the regard with which that bank was held.  These banks did not run without reserve, for the reasons given about, but they were free to choose how much and how to do it.  The only reserve requirement was having enough to convince your customers to use you...

A more modern example is "points".  Plenty of businesses award "points" for various activities, which can be redeemed for goods and services.  If you think about it, these points are like a constrained version of exactly your situations.  Usually points can only be exchanged with the issuer, but if you added the possibility to transfer points from one person to another, then you have a basic currency operating as you described.  The key feature here, is that these points are not backed by anything, and cannot be converted to gold or "hard currency", only airline flight, or groceries or whatnot.