I'm a huge effects junkie, I love how stepping on a box instantly can change tone and inspire one's work. THAT being said, my bare bones board is a tuner, compressor, octave, overdrive/DI (think ToneHammer or MXR M80) in that order.
Tuner - Must have for a gigging musician. Also doubles as a mute which is super helpful.
Compressor - Compressors are interesting because they can go from squeaky clean to rather "dirty" and/or squishy. A lot of a bassist's tone can be influenced by their compressor, in both a positive and negative way. Check out the difference between the MXR M87 and the Fairfield Accountant...you'll see what I mean. I'll go out here and say that more subtle compression is definitely more noticeable in the studio, but live, heavier compression and peak limiting makes playing significantly easier when switching between a pick/fingers/slap and taming peaky effects. It's just a good tool to have on standby and you should definitely know how to use one.
Octave - I play 5 and 6 strings pretty regularly, but nothing beats the beefy subs of an analog octave. It can change your tone rather dramatically and adds a lot to a song. Honestly, I consider it an essential given the versatility of it. It can be used during climatic parts to fill out the wall of sound, solo'd to create a sub/synthy bass or used in conjunction with fuzz and filters to emulate an analog synth.
OD/DI/EQ - It's just the perfect tone shaper. There's a reason why Sansamps are so popular, and its because they afford a bassist SO much flexibility in such a small package. I use mine essentially as an amp head, but also as an active EQ for passive basses. The OD can be always on, or kicked in for some grind, or set as a clean boost. See where I'm going with this? Plus, you can send your sound straight to the board.
All other effects are just gravy, and for those I'd definitely look into a decent multi. Find out what you like, and dedicate the time and money into those specific effects. Sorry if this was wordy, and enjoy the descent down the rabbit hole!