Substitute a button for a timed event?
What's wrong with ICs now??  

Anyway, what you want is a astable multivibrator, which is often made with an LM555 (IC!), but you can also make it with a pair of discrete transistors:  



The period T = time the button is pressed (t1) + time between button presses (t2), where  

t1 = 0.69 \$\times\$ C1 \$\times\$ R3, and
t2 = 0.69 \$\times\$ C2 \$\times\$ R2  

Use the output to drive a reed relay via another transistor. Place contacts of the reed relay parallel to your button. If a contact resistance of a few tens of ohms is OK (the reed relay will be milli-ohms) you can use a 74HC4066 as the switch. But then we're again using an IC...