You state that it's fairly simple. It isn't...

#Tight coupling Everything in your code is tightly coupled. Even the id of the element where the time is displayed is hard coded into the code.

What if I want multiple timers next to each other? Useless.

#Strange naming "convention" You use the word offset alot. But offset to what? What is your default time? After digging around I think it is UTC?

Then there is your method that let's you change the way the time is presented: setOffset. It accepts 2 arguments. raw and dst. What the hell is raw and dst? Again I have to dig into the code and still have no idea what you are doing. I think that raw and dst have to be passed in as seconds? But what is the difference between those variables? I have no idea.

dst seems to be optional. why? No idea. In the end you always count dst and raw together everytime you use them.

#Some other strange doings you create an IIFE (immediately invoked function expression) but you give it a name (function increnment(){})();. What this piece of code does is create function called increment. It then immediately invokes that function. The return value of that invoked function is stored in LiveTime.timeoutId. Or no, wait. The invoked function itself writes to LiveTime.timeoutId. Wow, do I need to explain more?

Not to talk about the horror of increment() actually caling itself. Inception!

#KISS: Keep It Stupid Simple Every function of your code should do 1 thing and 1 thing only. And preferably it should solve a little problem.

#Defining the problem The first step in solving a problem - code should always solve a problem. If it doesn't, you don't need it - is defining the problem.

Your problem:

I wanted to be able to automatically change the time and not deal with the hassle of local client times.

Actually consists of 2 problems (note the and in your sentence). The first can be solved by using timeouts, or even better an interval. The second should have it's own method that then would be called by the interval function. This method would take care of returning the current data as UTC (for instance).

You cram everything in an thing that looks like an IIFE whos return value is stored in a variable (but evnetually the function returns nothing itself, but sets the value in the IIFE itself).

You state that it's fairly simple. It isn't...

Tight coupling

Everything in your code is tightly coupled. Even the id of the element where the time is displayed is hard coded into the code.

What if I want multiple timers next to each other? Useless.

Strange naming "convention"

You use the word offset alot. But offset to what? What is your default time? After digging around I think it is UTC?

Then there is your method that let's you change the way the time is presented: setOffset. It accepts 2 arguments. raw and dst. What the hell is raw and dst? Again I have to dig into the code and still have no idea what you are doing. I think that raw and dst have to be passed in as seconds? But what is the difference between those variables? I have no idea.

dst seems to be optional. why? No idea. In the end you always count dst and raw together everytime you use them.

Some other strange doings

you create an IIFE (immediately invoked function expression) but you give it a name (function increnment(){})();. What this piece of code does is create function called increment. It then immediately invokes that function. The return value of that invoked function is stored in LiveTime.timeoutId. Or no, wait. The invoked function itself writes to LiveTime.timeoutId. Wow, do I need to explain more?

Not to talk about the horror of increment() actually caling itself. Inception!

KISS: Keep It Stupid Simple

Every function of your code should do 1 thing and 1 thing only. And preferably it should solve a little problem.

Defining the problem

The first step in solving a problem - code should always solve a problem. If it doesn't, you don't need it - is defining the problem.

Your problem:

I wanted to be able to automatically change the time and not deal with the hassle of local client times.

Actually consists of 2 problems (note the and in your sentence). The first can be solved by using timeouts, or even better an interval. The second should have it's own method that then would be called by the interval function. This method would take care of returning the current data as UTC (for instance).

You cram everything in an thing that looks like an IIFE whos return value is stored in a variable (but evnetually the function returns nothing itself, but sets the value in the IIFE itself).