Games for ChromeBook [Developers Mode not required]
Not many games can be played on the Chrome 42 update without developers mode since google disabled java, but after poking around my browser for a while, I found 2 sites to play some decent games that don't require developers mode/java. Below, 2 sites are listed and in order, there will be instruction on how to use those 2 websites.
How to use R2 Games
Wha R2 Games is, is a website that contains many RPG games for your PC(Keep in mind that these games do not require Unity, Java, Linix or any of the sort)
1. So, the first step your going to have to take is to click on the link provided above
2. Then you have to click on the register button shown in the diagram below

3. After you have done that, register with you personal information(you also have
the choice to register via facebook[as shown below])

How to use PlayCanvas
PlayCanvas basically brings WebGL games(What these are is basically Unity, Java supported games just not as good) to any OS you use or any browser you use.
1. The first step to using Play canvas is to click on the 'Sign Up' button in the top right corner of the page.

2. Now, after you have clicked on the button, you should get a screen like the one below.

3. You can just use your email and password or you have the choice of signing up with Google plus or Mozilla Persona.
So there you have it! Two really simple ways to play some decent games on your ChromeBook without the need of develepors mode (ecpecially if you ChromeBook is linked with a school and they don't let you enable developers mode).