> I wonder what the fastest possible chemically-propelled-rocket probe is?
Slower than a Nuclear-ion probe. Nuclear in this case means a small nuclear reactor, say in the 1 MW power range. Plasma thrusters have an exhaust velocity of ~ 50 km/s, and it is reasonable to reach 3x exhaust velocity, thus 150 km/s. The mass ratio (propellant to empty mass) would be 20:1 in that case. For any kind of chemical rocket to reach that velocity, it would need a mass ratio of 10 trillion, which is seriously impractical.
150 km/s = 31.6 AU/year, therefore missions to around 300 AU would be reasonable (10 year trip time). 1 MW reactor with radiators would mass ~ 20 tons. 300 AU probe would mass ~ 5 tons. Propellant load would be 25x20 = 500 tons. Propellant flow rate is .57 grams/sec or 49 kg/day. So thrust time is 28 years, which is a bit long. It would help if the reactor could be made lighter.