Clash Royale – What Can Gems Be Spent On?

EDIT: Following the recent update as of 03/05/2016 the current chest cycle can be found HERE

Original Answer: The comments are right, and apparently also is the reddit post.

It is pre-determined when you get a drop by this updated list (to include new chests). I just got a giant, so I know exactly where in the list I am, and can start logging from there. Few things to note:

1. Having no free space, but wining a battle, moves you to the next chest. So In theory you can always get golden and above.

2. The list might get updated or the entire system scrapped by Supercell at any time, given they are the developers and can do whatever they please.
3. Having gone trough the list, it starts over from 9.

A win would push you forward in the cycle, regardless if you have a free slot or not. That's a win where trophies exchange place (eg. not against clan-member in a friendly match)

EDIT: What I've just red on this FAQ is having full chest slots and wining does not progress you on the list

As a free-to-play player, there are two main things you can do with your gems to conserve it; make it last longer:

• Speed up chests that are nearly completed, not those that have just started.
  Alternatively, don't use them at all. You should be trying to save it for when cards begin to climb in costs (1,000 Gold at Level 8 to 9) and then [see below]:

Or, to get the most out of it:

• Spend your gems only on gold.
  Alternatively, wait until you have 500 Gems and buy the 10,000 Gold pack, and don't buy cards from the store. It is a known fact that you get more cards than gold.
• Never spend your gems on purchasing chests from the store. It's not worth it.
  That is, unless you're trying to get guaranteed Epics or Legendary cards, which you shouldn't be doing until most of your cards are maxed as the cards and gold values don't add up, even without the gem prices scaling with your Arena number.
  One exception to this are the occasional Epic Chest (using gold) and Legendary Chest (500 gems), which are much more profitable than other chests available for gems, or cards for gold.