Also note that once the maximum number of Bitcoins are created (~21 million) fees will become more important to the network as the miner will no longer receive new Bitcoins as a reward.
Is it possible to increase the amount of bitcoins?
Not without a hard fork. But there is no point. Bitcoin is deflationary by design.
When it reaches the maximum number of bitcoins, could we assume that the fee will be increasingly expensive?
I don't think the per-transaction fee will really change that much, but the number of transactions will hopefully be much higher and so there will be more fees for miners to earn.