tuner, compressor, preamp/DI
I would add to this that you should strongly consider owning your own studio/stage DI box, like a Countryman Type 85/Type 10 or Radial Engineering JDI/J48. In most cases, the venue or studio will be able to provide you with a direct box, but it doesn't hurt to always have your own that you prefer and trust.
After this, you should also strongly consider an HPF, to help correct for rooms with poor acoustics, and lighten the load on your bass amp, if you use one. I would only consider other EQ if there is a specific sound you are looking for that you can't get out of what you already have.
After that, lately, I have become enamoured of the Boss NS-1X Noise Suppressor. It's the first noise suppressor/noise gate I've ever heard that doesn't mangle my signal.
Beyond that, effects are a matter of personal taste. Always start with the utility gear.
To expand on my previous post, does anyone *really* "need" pedals? That's really just a matter of opinion. It is of course entirely possible to play an entire career without pedals, but pedals can and will do things for you that instrument, fingers, and amps cannot.
Most obviously, compressors, as someone else has already correctly pointed out, can shape your tone in ways that you as a player simply cannot achieve, no matter how much you practice or how "good" of a player you become.
It doesn't matter how good of a player you are, you cannot correct for a room or venue with poor acoustics without an EQ somewhere in you signal chain that is more extensive than the typical amplifier 3-band EQ or the tone controls on your bass.
And as far as playing without a tuner, well...that's a serious breach of professionalism, in my opinion. Serious musicians tune to a standard, not by ear, and it's no stretch to say that most musicians would agree.
We are spoiled for choice in the 2020s. Pedal effects today are better than a lot of studio rack gear of 20-30 years ago.
My answer doesn't change. If you are going to use pedals, I recommend the first three you buy should be a tuner, a preamp/DI, and a compressor, and the fourth should be an HPF. All of these are useful across all genres of music. Beyond those, it's really just a matter of what you think fits the music you are playing.
I have no use for an octaver or an auto wah (Mu-Tron), but some people like those. I can only think of one song I might want to cover that uses delay on bass, and while I did once apply chorus or maybe flanger to my bass in mixdown on one album as a special effect, I wouldn't bother with that live.
I started playing bass around about 1994, so I've been playing for 30 years now, and for most of that time, the only thing other than my amp and cable that I used was a tuner and sometimes a stage DI box (because all of my amps have had built-in DIs, I usually used those, unless whoever was running sound was insistent on using their box). First was a Boss TU-12P handheld, until that disappeared at a gig one night, then I got a TU-2 pedal.
My first amp head of my own was a Trace Elliot Series 6 AH200, so it had built-in graphic EQ and "Pre-Shape" EQ, but after I switched to a Trace Elliot V-Type, I also bought a Trace Elliot SM7EQ pedal and SMX Dual Band Compressor pedal. I rarely used them, and eventually sold them.
Later, I switched to a rack system, and had a JoeMeek VC1Qcs channel strip with its excellent optical compressor and an SWR Interstellar Overdrive preamp for dirt, but while that rig sounded amazing, it was just too much, so I went back to the V-Type and just my tuner, until about 2015-2016, when I started building my first bass pedalboard ever.
By this time, I has switched to my fretless StingRay with piezo as my main instrument, so my first real bass pedalboard consisted of tuner, compressor, acoustic resonance enhancer, and reverb. The empty space on this olivewood cutting board was where my Countryman Type 85 DI would sit.
All my bass pedalboards since have followed this pattern, but I have added drive and boost. My current main bass pedalboard is two vintage Boss BCB-3 boxes. That TU-2 is the oldest pedal I own. I'm pretty sure I bought it right after it hit the market, which the Internet says was 1998, four years before I joined TalkBass:
For the next version of my second bass pedalboard, intended for use as a direct to PA or recording console fly rig that will likely become my main bass pedalboard, I'm going with a mix of Tech21NYC and tc electronic, plus a Peterson tuner. This will add extensive EQ shaping for room correction (including HPF) and a looper (for practice only, not performance):
If my pedalboard were to stop functioning or be lost/stolen, I would be absolutely fine with plugging straight into the amp or PA or recording desk, if need be, with nothing more than my Countryman Type 85 DI and my tuner, or even a borrowed tuner and a DI box from the venue or studio. My tone would be different, of course, but it's not like it would be a showstopper.
I also own a Boss GT-1000Core that in and of itself can replace my whole bass pedalboard, soup to nuts, but I've decided to dedicate that to my guitar pedalboard.