Bulking Up Our Emergency Fund

I will get my first paycheck in a few days and it will go directly into savings for our emergency fund.  Since we moved into our house, our emergency fund has hovered in the couple thousand dollar range when it should be sitting at the $10k+ range.  Without this cushion I haven't been financially comfortable because of all of the things that could happen that would create a financial disaster due to the fact that we have such an anemic emergency fund.  Among the disasters that would require more than a couple of thousand dollars to fix:

• Our one and only car gets wrecked (since we only carry liability on our old car, if the accident were our fault we would be minus our only car and would need more than $2000 to buy a suitable replacement).
• Our air conditioning system dies (we would be just fine if our heating system dies--it doesn't get that cold here and we could use space heaters--but air conditioning is A NECESSITY in Las Vegas.  And a new air conditioning unit could cost up to $5000!).
• One of the kids has a dire emergency (all of the kids are scattered around the US and one is in the Philippines.  If one were severely injured or had another critical emergency, getting to them, renting a car once we got there, funds to help out their family, and other travel/helping expenses would cost way more than $2000).
• One of us has a dire emergency (when one person in the family has a dire emergency--like being severely injured or having a severe medical crisis--the average cost of living, even though our house/car/everything else is on hand, can still grow exponentially.  Fortunately we have full coverage medical insurance but additional costs--like driving to the hospital and back daily, eating out every meal, other extras incurred when one is hovering between home and hospital--can grow exponentially and blow through a tiny emergency fund in days).
• The government goes over the financial cliff (because we both rely on the hubby's income and because both of his monthly checks are issued by the government--Social Security and pension--one glitch in the government's financial system could leave us without an income for a month or longer.  Thus the need for a much bigger than average emergency fund).
• One of us dies (knocking on wood that this doesn't happen for a very very long time...but an unexpected death has many expenses including a funeral, cremation/burial, probate costs, etc).
• We get divorced (ditto the knocking on wood thing but I've seen way too many people surprised by their spouse filing for divorce and neither them nor their future ex have any cash on hand and things go down hill from there.  If one or both or the parties don't want to end up homeless during this trying time, an emergency fund that is split by both is better than being penniless).
• A natural or man-made disaster strikes (you see all of those houses destroyed by wildfire in Colorado this week?  It made me think about what would happen if we were impacted by a similar disaster.  We would need to pay for a hotel or other place to stay, we would need to replace clothes and toiletries, etc.  Insurance would come into play eventually but the moment a disaster happens is when you need immediate cash on hand).

That's all of the terrible things I can think of now but you get my point.  Without a fat and fluffy emergency fund, a major crisis can have a majorly bad impact on your life.  That's why all of my hard work is going towards bulking up our emergency fund instead of to something more fun...like shoes :)