My solution requires the values of the selection lists to be in a specific format.
If the parent selection lists has the options
1
2
3

The "Child" selection list's options must start with the relevant parent option plus a hyphen e.g.

1-A
1-B
2-A
2-C
3-D

Selecting "1" in the parent dropdown will filter the child dropdown so only options starting with "1-" are available.

Put the following in your custom/js folder: https://gist.github.com/JohnKingsbury/6e103d3e9f2b865c431ccfbb4fcb8842

And in the customcontent of a screen, it can be used like

crm.ready(function(){
  filterSelect("comp_c_myparent", "comp_c_mychild");
})

Here is the one I use, this goes into the Custom Content of the page 

And works on the Code in the first field being something like A, B, C and the second field A1, A2, B1, C1. The coding matches the letters 

<script language="javascript">
crm.ready(function(){
FilterSelection("#case_field1","#case_field2");
FilterSelection("#case_field2","#case_field3");
FilterSelection("#case_field3","#case_field4");
//add as many as you want here

});

function FilterSelection(main, linked)
{
var list = $(linked).html();
$(main).change(function(e) {
var selected = $(main).val();
$(linked).html(list);
if (selected == "--None--") return;
$(linked).children("option:not([value^='"+selected+"'])").remove();
$(linked).append("<option value='' selected=true>--None--</option>");

});
}

</script>

Sweet thank you Matthew, this has got me going in the right directions. However there is a small issue. When first coming to this page this doesn't automatically set those values as it only runs when I change the original field. What is the method for setting the available values when first coming to this page and clicking the Change button

I think (as my dev gave me the script) you just need to call it, so provide it a name (so change function() to be function(NewName)) and then you can call on this from the CreateScript of the main fields 

How do you deal with situations where I have a dropdown list where some of the options appear for more than one option in the parent list? For example:

Parent list:
A
B
C
D

If I choose A, I need options 1, 2, and 3 shown in the child list. If I choose B, I need options 2, 3, and 4 to appear. In this example choosing either A or B, both will have the common elements 2 and 3 shown in the child plus their additional unique ones.

My original script would require you to make multiple sub-values 

Another way to do this is to approach this differently. Ignoring my script you could do it this way. Lets say we have two Selection fields 

Selection-A (e.g. A,B,C,D)

Selection-B (e.g. 1,2,3,4,5,6,7,8,8,9,10)

And we want A to control B. But you want values in B to be available to multiple values. in A. 

Create a 'Master' B list in the normal fashion 

Then within Translations you manually create an entire new list of items (e,g, capt_family='Selection-B1') which only has (1,2,3,4) and then do it again for capt_family='Selection-B2. Which has (1,3,5,7) 

Then on your screen within CRM you could put an OnChange script in on the Selection-A field that If Selection-A = A then change the capt_family of Selection-B to be Selection-B1 but if you change it to B then change it to Selection-B2

So long when you build the values you use the same codes in B1 and B2 just choosing to not include them all. 

Now that is thinking out side the box Mr. Shaw, I utilized something similar to what you said however I just hard coded each value in my code. Thanks again in time switching to your recommendation pertaining to Capt_family may very well happen.

Now that is thinking out side the box Mr. Shaw, I utilized something similar to what you said however I just hard coded each value in my code. Thanks again in time switching to your recommendation pertaining to Capt_family may very well happen.

I've implemented it once at a customer site with just two selection fields, but built 4 custom capt_familys for the sub level, but works a treat. It is a headache for maintenance but so long you're not needing to add values all the time it should be OK.