Is it possible to have vinyl nearly noise free?

You don’t mention what your system is. I think every element of the system has something to contribute to this from the quality of the LP to the cleaning process, the cartridge, the turntable and motor, and the phono stage.

In my system, noise is almost nonexistent or at least so minor that it is not an issue for me so here is my system and approach.

1) I buy only LP’s that are rated VG+ or better from dealers with a good reputation
2) I clean before each play on a Loricraft cleaner
3) I have a Transrotor Fat Bob TT which has magnetic floating bearing (IMHO this is the single most significant factor in reducing noise. I have had several other turntables and none have been as quiet as this)
4) I have a ZYX Airy 3 cartridge and Graham Phantom 2 arm

There are certainly better systems out there but all I can tell you is that with mine, clicks and pops or any other noises are a non-issue.

I have found LAST products excellent. 

PAY ATTENTION TO ATMASPHERE! He explained why my new phono preamp lowered (pretty much eliminated) LP playback noise.

In 2012 an audiobuddy sold me his MAGI (Massachusetts Audio Group, Inc.) line stage. MAGI is pretty much a one-man operation and they don’t come up for sale often. It made EVERYTHING sound better, because all components go through the line stage.

Two years later, he asked me if I’d be interested in MAGI’s matching phono stage, the aptly-named Phonomenal. Both components are handwired PTP with NOS tubes.I think I laid a strip of rubber from my driveway to his house. I gave the Phonomenal a careful listen and bought it on the spot.

When I got home, I started playing records through it and shook my head in disbelief at the absence of surface munge and ticks and pops.

I posted my experience to this forum to see how the lack of noise could be possible. Atmasphere responded and explained that many phono stages use *active* negative feedback to reduce measurable distortion, but as the signal is recycled through the feedback loop, it lengthens the duration of the ticks and pops, making them louder and longer than they occur on the record surface. Passive EQ passes the signals along without lengthening the noise.

It made sense to me, and five years later I am still enjoying the hell out of my vinyl rig, with mostly noise-free playback and none of the obsessive scrubbing and cleaning some of you describe.

AND! If you have a similar rig and REALLY want no-noise LP playback, get some mono albums and play them back through a mono cartridge.

Mono albums’ grooves modulate only side-to-side, meaning that true mono cartridges only respond to side-to-side groove modulations. I have gotten many used mono LPs that were too noisy to play with stereo cartridges, that were noise-free when played with mono cartridges. I have a relatively inexpensive Audio Technica HOMC mono cartridge that plays mono albums sweetly and noise-free, even 1969 albums plucked from the bins of thrift shops. My Beatles 2014 Mono LP remaster/reissues sound phenomenal with mono playback.

@voiceofvinyl & @atmasphere - Nice posts, thanks!   I have a few LPs that are dead quiet and one that is dead quiet except for a small scratch that is as loud as a gunshot.   You know, it only takes one LP to prove you can have quiet LP's, one just doesn't make it the norm.  Unfortunately.

Fleshler I always avoided playing an LP twice in a day as the prevailing thinking was as you describe. It is true the vinyl under the stylus is under immense pressure and is described as going liquid (never proven as far as I can tell). The mass of vinyl involved is so small that the surrounding vinyl acts as a heat sink and the temperature returns to normal immediately. I have a tire infrared temperature gauge and can not detect any change in the temperature of the record while playing. I think as long as the record and stylus are clean it is safe to play a record over again if you have too. But who wants to listen to a record twice in a day;-) 
"Last" is a joke. It is nothing but freon. You can prove this to yourself easily. With a dropper place a drop of Last on a glass slide and let it evaporate. Put another drop in the same place on the slide and let it evaporate. You can keep doing this a thousand times over and you will never see anything remaining on the slide. Nothing, Nada, Zero. That is how gullible we are. Will freon change the vinyl in any way? Absolutely not. It is totally inert which is one of the reasons it is a great refrigerant.
Lps are Lps. Some are quiet, some not. The best way to keep a quiet collection is do not buy used records. There are exceptions for instance if you are buying the estate of an expired audiophile who had so many records he may have played one once a decade. Make sure he did not buy used records.