Because I work as a government employee, I can't earn miles if there is a cost associated with getting them. Therefore, I don't earn miles at car rental companies which charge "Frequent Flier Miles Surcharges" or whatever they're calling them these days. (Such a waste...) Are there any rental companies which don't charge a fee for earning points or miles?

Take the hotel or foreign airline points instead.

While this may be true, you may have another option:

Instead of earning airline miles from a car rental program, how about earning free days of car rentals instead?

Gleaning from personal experience, I primarily rent from National Car Rental as I am an Emerald Club Executive member. When I recently switched my preference from earning airline miles to earning free car rental days, the fee for earning frequent flyer miles was subsequently dropped.

I believe other car rental companies have a similar policy.

Please refer to the Earn Points in Rental Car Programs for Free Car Rental Days! thread for more information.

Agree with both of the above posts. Pay close attention to your favorite company's home-grown points program (e.g., Hertz #1 Awards, National's revamped Emerald program, etc.) as well as the opportunities to earn hotel points for your rentals. Based on your own personal value and need for car rental or hotel points, you can make the best decision for each rental.

Hotel points are typically a fixed amount - perfect for short, cheap rentals. The rental points (at least at Hertz and Budget) are awarded per-dollar, making them perfect for longer or more expensive rentals. The breakeven point varies based on your own valuations, but for me it's about a $70 rental (More expensive, and I opt for Hertz ZE1 points. Cheaper, I take Marriott points.)

I usually like Qantas for short rentals: 700 points per non-Australian rental.
Amtrak gives 50 points per day & has a 500-point bonus PC with Hertz, both of which are Doubled in Double Take until June 30.
I also like whichever hotel program has the best bonus that month .

I've been collecting Virgin points. I don't know what I'm going to do with them but I can always turn them into more Hilton points. They're also free of that silly tax.

About a year or so ago with Hertz, I tried to switch from miles to #1 award points and found that I was not allowed to receive #1 award points if I used the CDP from my government organization (i.e, government rates did not qualify for earning #1 award points).