How can I check tire pressure without a pressure gauge?

The best "heuristic" is to observe the rear tire under full weight conditions. It should deform noticeably, but not to an extreme. Deformation should be roughly 10-20% of tire height from ground to rim edge -- less deformation for road conditions, more for off-road.

Lean most of your weight onto the seat or handlebars (depending on which wheel you're trying to check) and watch how much the tire bulges. Do this on a road-like surface, not dirt or carpet. Generally the tire should bulge out a little next to the contact patch, but not much. Try bouncing your weight, no matter how vigorously you bounce the tire shouldn't get close to bottoming out. There should be at least a little bulging, especially when you bounce hard.

This works better, of course, if you've used a gauge with those tires before and know how much bulging to expect with proper inflation.

It's just too hard to judge with your fingers. Odds are good that a little under inflated and correct inflation both feel almost rock hard.

If you use a hand pump and know it well, you can get a rough gauge from how much the resistance increases as you pump. I find that gets me within 10-15 PSI when checking later.

Get a pressure gauge. They're not expensive (I had one of these cheap plasticky ones as a kid. Not the easiest to use without losing air, and no good for racing tyre pressures, but served me fine for many years. These days I use a pump with a built-in gauge.)

Heuristics are unreliable, especially if you're not already used to how your tyres look and feel at the correct pressure. Inflating your tyres sufficiently is key for reducing punctures and decreasing rolling resistance, while overinflating them will give a harsh ride and could eventually be dangerous. Why take the risk?

The most important thing is to have a pump with a pressure gauge. Just top off the tire pressure on each tire before each ride. You can then squeeze the tire and over time get a feel for what is the right pressure for you.

The "right pressure" depends on your weight and what type of tire you're dealing with, and your preferences. And the only way to determine this is to ride your tires at different pressures and see what feels right. It is somewhat subjective. I DO NOT think that 2-3 psi makes much difference, and I know that some people will vehemently disagree with this.

Generally speaking (to give you a starting point for experimentation): If you're a "clydesdale" (over 200lbs), you'll want to be above the recommended pressure that is printed on the tire. If you're fairly lightweight the recommended pressure is a good starting point.

Another thing to try is to run your front tire at a lower pressure than the rear. This will dampen the road shock to your hands. Again, how much lower is dependent on experimentation.

Some people have more rigid rules and heuristics than me but if I had to pick one set of rules I'd choose these from rivendell, they describe what type of tire and tire pressure is appropriate depending on rider+bike+usage.

You can indeed check by hand, however you will only be accurate if you practice it often and cross-check with a tire gauge whenever possible. Realistically you should always use a tire gauge if possible--however knowing how to check by hand is a good backup when changing a flat, or when away from home and in need of a quick check.

As others have said, buy a gauge! A good digital presta/schrader gauge can be bought online for roughly $25 US. Digital gauges are more consistenly accurate, however a less expensive analog gauge is still much better than guessing by hand.

Now, once you have a gauge, use it after every long ride on your road bike--or at least weekly (road tires lose pressure quickly, so daily is better yet). But! Before putting your fancy new gauge on, give your tire a squeeze and push the tip of your thumb into the side of the tire. Make a guess if the tires are "ok", "low" or "really low". Follow up with your gauge, add air if needed, then repeat the squeeze to see how they should feel at pressure.

Since there are 2 wheels, this bike tire needs to support at least half your weight, all the time. Double your weight, well it needs to at least take that force periodically as you hit bumps and curves. So it's safe to say you should test by putting all your weight on one tire at least for a sec, and if you used a hand pump it took quite a lot of strength to get it there. The worst thing you can do is overinflate it using a compressor because from that disaster, there is no going back or "undo" on that. I have seen one explode recently, minutes after getting why wheel home from the careless bike shop that boasted he didn't use measurement. It will start to bulge out the side, and then within seconds, explode as violently as a hard slap, literally in your face as you race to get to the valve to let some air out. In my case as in probably all cases, it blows out the side of the tire and so the inner tube and tire both must be replaced new.

Next time you have a gauge you can inflate more to the max, take a reading and compare with how it deforms while learning to eyeball it better in the future. And no, I don't go back to this bike shop when I told him what happened because he tried to blame the tire which was brand new, and he didn't offer to even apologize about his staff not reading the low 60 max psi reading on the mountain bike soft tire to avoid overinflating. I didn't go blind or get a face or hand injury, but I had bought 3 tubes and was lucky to find a matching tire for a reasonable price.

If you are machinists you probably know about the “bouncing steel ball” material hardness test. The harder the material under test the higher the ball rebound towards its start altitude, always less than 100%. To your question .. The same principle operate in a reverse order to let it test how hard our wheel (ball) is instead of hardness of the concrete floor. Same principle applies to a basket ball. The catch is we need a standard for reference - ie our bike tire inflated to a target value. Measure how high (peak cm/inch) in first rebound and take it as a nominal record. We can, if we like, to also create a lower limit rebound record that will tell us if we need to inflate or not.