Earl commented about 3 hours ago new Just looking at your puzzles make my brain cell hurt. I gotta get a CNC or laser. I could make so many intricate pieces.
CNCs are great as they can do 3D which to me is a ++, however, they toss pieces around, especially small ones that are hard to anchor down... I believe the better you anchor them down, the harder they are to extricate. Also the sawdust is a pain... just imagine a router working 100 times faster than you can. Friends have spent more shekels on dust collection that the actual machine... but that's just hear say.
As far as I'm concerned, the main issue with the laser is smoke fumes... especially MDF, though many people avoid it and tend to go for ply as an alternative (or thinner solids). I use MDF not so much as cost, but rather availability. Even C grade ply costs a 3rd mortgage in my locality.
If you do go laser, ensure you can cut at least 3mm, but preferably 6mm and anything above is just icing on the cake.
To get thickness, I use laser cut holes (2mm, 4mm or 6mm) for alignment and have laminated to thickness of 75+mm, here is a good example, so if you can do 3mm, all it'll cost is a shitload of glue and dowels.
I think the main beauty of CNC and lasers is repeatability... hell most of us strive for that and spend fortunes on Incra and 'peckers to achieve that.
Get more inspiration from Splinter... he is spearheading what some may consider the budget spectrum of lasering and what he has achieved/demonstrated is inspirational (I expect a BIG cheque for that Splint, before they are obsoleted).
If your first cut is too short... Take the second cut from the longer end... LBD