Boosting volume after the fact

If you have an LUFS meter at the end of your master bus FX, then when you bounce and measure, it should be the same, unless you've messed with or automated (or "bumped" - stuff happens...) your main/stereo-out fader. I think I paid for the full version of Youlean just to have a standalone I can drop audio or video files into and confirm the final result is where I want it, i.e., before uploading to SoundCloud, YouTube, or sending to someone.
To answer the original question, normalizing finds the biggest peak and brings that to a certain level. If there is one peak in the entire song that is way louder than the rest, it will stop the volume of the sound from getting any louder.

If there was an app that did get it louder, it had to knock down that peak to bring up the rest of the track. Most likely using a limiter.
Above, I mentioned that you need to look at what the program/plug-in is doing. Here's an example. I took one of my tracks that measures -18.5LUFS with a max peak of 0.05dBFS which is just past half way. I plugged the file into Audacity, and use the Loudness Normalization function, set the value at -14LUFS. Here are the results.
Audacity Loudness Norm.jpg
Clearly you can see that the program didn't use any compression or limiting in the process, just calculated how much it needed to increase the volume and cranked up the gain until it got a -14.0LUFS. Consequently, there are lots of peaks that are completely flat topped.

A better sounding process was to put a compressor on with a compression of a 2.4 on peaks above -10dB, and use make up gain. This gave me a LUFS of -14.2, sounds as loud as the Loudness Norm function but there are spots where it's not quite as spitty sounding, especially on snare hits. This is the same section, you can see that there are no flat topped peaks. It does have a bit more compressed sound vs the original, but it is sounds a bunch louder.
So it's important to know what the process is doing to the file.
I should have thought to do this earlier... Here's the original MP3 rendered at a lower than usual volume level. Then the same MP3 processed through a 3rd party app to boost volume - volume increased by 3dB.

The boosted MP3 shows clipping in the first half measure.


Then it smooths out for the rest of the song.

Right Spantini, it looks like the very first peak is maxed out which will set the normalization level.

You can see the flat topped peaks pretty easily. As to whether you can hear them is another matter. It it's two peaks out of a 3 minute track, I doubt that anyone would even notice!
The best way that I've found to automatically normalise tracks to the same level is to use the SWS Loudness tools. The SWS extensions are a vital addition to Reaper and I couldn't work without them these days. The Loudness tools allows you to analyse tracks (I would suggest using it on finished mixes because it doesn't make much sense using it on individual tracks) and then normalise all the songs so that they all play at roughly the same level according to the LUFS reading. If you are being really fussy you may want to manually adjust the level by one or two dB after running the normalise but normalise quickly gets you into the right ball park.

I tend to use a low value for LUFS and then make up the gain with ReaLimit to get the desired level.