Assuming your investments aren't in any kind of tax-advantaged account (like an IRA), they are generally not tracked and you indeed may pay more taxes. What will likely change, however, is your cost basis. You only pay tax on the difference between the value of the investment when you sell it and its value when you bought it.
There is no rule that says once you sell an investment and pay taxes on the gain, you will never again pay any taxes on any other investments you then buy with that money. If you own some investment, and it increases in value, and then you sell it, you had a capital gain and owe taxes (depending on your tax bracket, etc.). If you use the money to buy some other investment, and that increases in value, and then you sell it, you had another separate capital gain and again owe taxes.
However, every time you sell, you only are subject to capital gains taxes on the gain, not the entire sale price. The value of the investment at the time you bought it is the cost basis. When you sell, you take the sale price and subtract the cost basis to find your gain,
So suppose you bought $1000 worth of some ETF many years ago. It went up to $2000 and you sold it. You have $2000 in cash, but $1000 of that is your original money back, so your capital gain is $1000 and that is the amount on which you owe (or may owe) taxes. Suppose you pay 15% tax on this, as you suggest; that is $150, leaving you with $1850. Now suppose you buy another ETF with that. Your cost basis is now $1850. Suppose the investment now increases in value again to $2000. This time when you sell, you still have $2000 in cash, but this is now only $150 more than you paid, so you only owe capital gains taxes on that $150. (A 15% tax on that would be $22.50.)
In that example you had one capital gain of $1000 and a second of $150 and paid a total of $172.50 in taxes (150 + 22.50). Suppose instead that you had held the original investment and it had increased in value to $2150 and you had then sold it. You would have a single capital gain of $1150 (2150 minus the original 1000 you paid). 15% of this would be the same $172.50 you paid under the other arrangement. So in essence you pay the same taxes either way. (This example is simplified, of course; in reality, the rate you pay depends on your overall income, so you could pay more if you sell a lot in a single year, since it could push you into a higher tax bracket.)
So none of the money is "tax exempt", but each time you sell, you "reset the counter" by paying tax on your gain, and each time you buy, you start a new counter on the basis of whatever you pay for the investment. Assuming you're dealing with ordinary investment instruments like stocks and ETFs, this basis information is typically tracked by the bank or brokerage where you buy and sell them. Technically speaking it is your responsibility to track and report this when you sell an investment, and if you do complicated things like transfer securities from one brokerage to another you may have to do that yourself. In general, however, your bank/brokerage will keep track of cost basis information for you.