A simple, analog way to do this would be to use an LM3914 Dot/Bar Display driver.
Set the 3914 is set to operate in the Dot mode (only 1 output lit at a time).
The SIG input to the 3914 is a resistor with a capacitor to ground to provide a ramp voltage when power is applied. As the input ramps up the outputs will sequentially turn on.
Connect the LED in parallel to every other output (i.e. connections for LED 1,3,5,7,9,10) which will cause the LED to turn on and off 5 times and then say lit at the end.
The time constant of the input RC is selected to give the desired flashing time (R x C probably around 1 second).
The voltage to the RC input should be the same as the RHI voltage (typically REF OUT). Thus at the end of the input ramp the last LED driver should stay on and keep the LED lit.
Because the input is a exponential ramp from the RC circuit the LED blinking will be more rapid at the start then at the end, but I would think that would be acceptable. If you want a uniform blink rate then you would need to add a constant current source to charge the capacitor, which would give a linear ramp, and a constant blink rate.
The circuit requires 1 capacitor and 2 resistors in addition to the LM3914. The 3914 has a built-in constant current drive for the LED.