Yeah sure you can do this, depending what you define as a small amount.
On Poloniex for example, you can deposit as little BTC or altcoins as you like and start trading, they don't seem to have any minimum deposit requirements listed.
How much do you have to start trading with? If it's just dust it's obviously not going to be enough for the fees etc.
the OP should have pit how much he want to start investing because if
thats just a dust i advice him to just start ranking his account and join campaigns
so he could earn capital for trading.but if he has enough amount to start,so Mate
is right poloniex has no wminimum amount to start investing so better check
them out for details