Thanks, Devendra, it’s a move in the right direction!

There were no validation rules on the custom object; however, I looked at the page layout for the object and found 7 fields that were required on the layout. 

So after adding values for those 7 fields, the code changed to:

        /* Create 1 test recurring donation */
        npe03__Recurring_Donation__c a = new npe03__Recurring_Donation__c(name  = 'testRecurringDonation',npe03__Contact__c = '003J0000006W3j4',RecordTypeId = '012A0000000qshA',npe03__Date_Established__c
 = system.today(), npe03__Amount__c = 1,npe03__Installment_Period__c = 'Monthly', npe03__Schedule_Type__c = 'Divide By',npe03__Installments__c = 12,Designation_Type__c = 'Unrestricted'
);
        insert a;

 and when I ran the test, I got a different error:

ChatterOpptyOnRecurringDonation: execution of AfterInsert

caused by: System.SObjectException: Invalid field npe03__Recurring_Donation__c.id for Opportunity

Class.ChatterUpHelper.chatterUp: line 63, column 1
Trigger.ChatterOpptyOnRecurringDonation: line 36, column 1: []

For this forum, would the etiquette be to drop this post here, and start a new one for the different error, or continue?  (I can modify code, but I'm not an Apex programmer per say.  Still hammering at trying to figure out myself)

In case of continuing, here's the code, I'm honestly not sure if the problem is the first or second reference to a.id or npe03__Recurring_Donation__c.id:

@isTest
private class testChatterOpptyOnRecurringDonations {
    static testMethod void testChatterOppty() {
        List<Opportunity> myOpportunities = new List<Opportunity>();
        Chatter_Up__c settings = createChatterUpConfig();
        insert settings;
        
        /* Create 1 test recurring donation */
        npe03__Recurring_Donation__c a = new npe03__Recurring_Donation__c(name  = 'testRecurringDonation',npe03__Contact__c = '003J0000006W3j4',RecordTypeId = '012A0000000qshA',npe03__Date_Established__c
 = system.today(), npe03__Amount__c = 1,npe03__Installment_Period__c = 'Monthly', npe03__Schedule_Type__c = 'Divide By',npe03__Installments__c = 12,Designation_Type__c = 'Unrestricted'
);
        insert a;
        
        /* Create 100 test opportunities */
        for(Integer i=0; i < 100; i++){
            Opportunity o = new Opportunity(Name = 'TestOpp_'+i);
            o.npe03__Recurring_Donation__c = a.id;
            o.CloseDate = date.Today()+i;
            o.StageName = 'Negotiation/Review';
            myOpportunities.add(o);
        }
        
        test.startTest();
        insert myOpportunities;
        test.stopTest();
        
        /*there should be 100 posts as you iterated 100 times*/
        List<npe03__Recurring_Donation__Feed> chatterPosts = [select id from npe03__Recurring_Donation__Feed where parentId = :a.id];
        System.assertEquals(chatterPosts.size(), 100);

    }
    
    static private Chatter_Up__c createChatterUpConfig() {
        
        Chatter_Up__c setting = new Chatter_Up__c();
        
        setting.name = 'Opportunity Settings';
        setting.object_api_name__c = 'Opportunity';
        setting.parent_id_field__c = 'npe03__Recurring_Donation__c.id';
        setting.chatter_post_body__c = 'This is a test';
        setting.chatter_link_title__c = 'This is test title';
        
        return setting;
    }     
}