Validation Rule on Multi-Select Picklist to only allow one value

What you're asking can't be accomplished through a validation rule, without hand coding every value like you suggested.

On the other hand this could very easily be accomplished with a before update trigger that looks something like this:

trigger MyTrigger on MyObject__c (before update)
{
    //this variable could also be gotten through use of the system describe methods
    Integer totalPicklistItems = 30;

    for(MyObject__c obj : Trigger.new)
    {
        if(obj.MultiSelectField__c != null && obj.MultiSelectField__c != '')
        {
            //split picklist into list
            String[] selectedValues = obj.MultiSelectField__c.split(';', 0);

            //check if the selected selected values are less than 1 or equal to the
            //total picklist items to add an error
            if(selectedValues.size() < 1 || selectedValues.size() == totalPicklistItems)
                obj.MultiSelectField__c.addError('You must select at least 1 value but not all values.');
        }
        else
        {
            obj.MultiSelectField__c.addError('You must select at least 1 value.');
        }
    }
}

If you want to return a validation error on a record that does not comply with your rules then you should wrap it inside an IF clause like this.

IF(
AND( 
$UserRole.Name == 'Career Advisor - Inbound', 
$Profile.Name == 'Acquire Sales Admin - Inbound',
AND(
ISPICKVAL(Lead_Sources_for_Career_Advisor__c, 'Career One'),
ISBLANK(Lead_Source_Referral__c)
)
), false, true)

This will give an error when this formula is not met.