Combining a partially miscible three-liquid system with interfacially trapped silica colloids, we show that small droplets can exhibit dramatic growth phenomena driven by physical effects alone. The mass dense droplets sprout tubes which grow vertically upward in a gravitational field and respond to the presence of other droplets in their path. Two of the liquids in our system are water and toluene. By varying the third liquid, we are able to relate the growth behavior to the details of the underlying three-fluid phase diagram and the changes to the interfacial tension. Additionally, we introduce a pendant drop in the path of our growing drop. We use this to confirm that growth is driven by the partitioning of solvents, that exchange of solvents between droplets is chemically selective, and that the exchange behavior can itself generate further growth phenomena.