Nearly every timed event in the game can be completed instantly using Drachmas. Before you go crazy spending all your hard earned Drachmas boosting everything around, understand that most boosts are quite expensive in relation to the advantage it gives you. However, there are quite a few circumstances where boosting an event may be advisable even if you are tight on your Drachmas.
You can not speed a unit on it's way to a target. If your troops travel at 18 MI(K)/H then you'll have to wait while the unit makes it's way to the target at that pace. However once your target has reached it's destination you can, anytime after that, instantly return them home. This goes for all combat missions and trades. You boost combat units in the War Council and Galleys inside the Port. You may chose to do this so you can send the next attack/trade out quicker but the biggest use of this is to pull your troops home early to put them in the Acropolis in the event someone decides to send an attack following your troops on their return trip. You don't need to spend large amounts of Drachmas to make use of this. Instead just wait until the last minute or two and then pull them home. It only takes seconds to put them in the Acropolis after they have been returned.
You may speed up the construction or upgrade of any building. This is especially useful if you need to instantly build up your acropolis against an incoming raid or if you are the Hegemon and wish to build up your storage capacity as quickly as possible for Coalition Upgrades.
Much like with buildings, you can instantly speed up the production of any Agreement. This can quickly give you access to a new unit that you just unlocked.
After queuing up the training of numerous units, you have the option of instantly completing their training. Be advised that this tends to be quite expensive, though it is certainly cheaper than buying the units outright via the Market. Unit boost can be triggered either in the training screen or from your city display (green arrows over the unit building). You also have the option to either boost the current set of units or all units being produced by the building.
You may optionally complete your Divine Quests instantly.