I have a Honda HRC215 lawnmower. It's a very nice machine, but became impossible to start.
The symptoms were that:
Over time it would get harder and harder to start, eventually becoming impossible to start without starting fluid. But starting fluid always worked, and once started, subsequent restarts were easy.
If the mower was left in the sun for a while before attempting to start, this greatly improved one's chances.
The first start of the year was very difficult.
I had followed the service manual, adjusting things as required. This would help for a while.
Long story short: It turns out that two things are happening at once.
First, fuel volatility. Gas that's been around for a while loses its volatile components to evaporation, making it hard to get initial ignition. Warming the mower helped because it raised the vapor pressure of what remained. What really helped was to replace the gas at the start of the season: pour the old gas into the car's fuel tank, and buy brand new gas. Note that the gas is not turning to varnish - the carburetor is clean is a whistle. A key clue is that starting fluid (ether) always worked on the first try.
Second, the choke cable. It turns out that the clamp at the motor end of the push-pull cable isn't secure, and slips slowly. I was trying to compensate by adjusting other things, and got deeper and deeper in trouble, as it drifted back out of adjustment. The net effect was that even when in starting position, the choke was not properly closed.
Now this system isn't especially complex, so I started looking for what was slipping. And the cable has exactly two ends, and the one at the control clearly could not slip. Which left the motor end. Turns out the clamp was the traditional kind, with a ridge that was supposed to drop into the V between adjacent turns of the coiled wire sheath, the positive metal-to-metal engagement preventing slippage.
Problem is that the coiled wire is square in cross-section, so there is no V, and the clamp was attempting to grab the soft plastic jacket. Which slowly flowed away, allowing the cable to move in the clamp, no matter how tightly it was clamped. By the time I looked at it, the jacket had become mushed and mangled, and and the cable would slip immediately. Clearly, a design flaw.
The solution was to make a retention clip of bent 0.0625" diameter music wire that goes under the existing retention clamp and prevents the cable from being pulled forward. This allows sufficient pressure to be applied that the choke is engaged.
Now, the mower starts right up.