If someone buys bitcoins on an exchange (assuming it's an exchange with KYC and you submit real docs), does that mean that your gov can now keep track of how many coins you have? Since every transaction is in the blockchain, they can just see that you bought 10 btc on bitstamp, then you transferred 10 btc to another address and then 10 btc from that address to another address, etc. So they will always know generally how many btc you are in possession of and what addresses they are on? Even if you split them between many different addresses, it will be obvious which addresses are throwaways based on their other transactions, and the btc can always be tracked.
Note: I'm not trying to do anything illegal, just interested in this topic, since "untraceability" is always something that bitcoin promoters talk about.
In theory, your story can happen if the exchange shares this information with the government (I doubt they do this by default, but KYC stuff can make them do it on request by the government).
However, there are ways to stop the trace. For example: I buy 10 btc on Bitstamp, send it to my personal wallet and then to any website that uses a shared wallet system. At irregular intervals I withdraw different fractions of my 10 btc from the site. Since they use a shared wallet, these withdrawals won't come from the deposit address I used. So here the trace stops. Anyone investigating the blockchain will see my deposit of 10 btc to the site and then my withdrawals are mixed in with everyone elses and since the addresses used are not linked to the address I originally had my coins on, there's no way to link them to me.
There are dedicated services that do just this, called mixers, but with a careful withdrawal plan, most shared-wallet sites work almost as well.
