Elliptical Workout Analysis

Hello Fellow Gear Heads!
When I started on the elliptical machine, I lasted all of five minutes. After a couple of weeks, I could last around 15 minutes before taking a break. I’m still on the beginner level (level one), but the plan is to max out the timer on the machine (it only goes up to 1 hour), and then start working my way up to level 20, however long that takes.
Because of finals, I had to let my workout fall to the wayside. I started working out again last weekend by riding hard on the drives around Fairmount Park (Forbidden, Kelly, Schuylkill, MLK, and a smidgen of Lincoln for good measure). I took Monday off, because the week’s workouts drained me. However, the next day I blew through twenty minutes on the elliptical machine as if it were nothing!

Six minutes went by before I even broke a sweat! I spun at 68 RPMs consistently with a steady heart rate of 164 BPMs. Around 17 minutes in I went up to 174 BPMs, which I didn’t think was too bad at the moment, so my first thought was that next week I’ll go to 30 minutes. Then I thought, “Why not do it now?” I didn’t even stop spinning. I added ten more minutes to the elliptical computer on the fly and kept humping.
At about 26.5 minutes in, I looked down and my heart rate was at 180 BPMs! I wanted to keep going, but I wasn’t sure if that was safe. I worried that I might hurt myself if I let my heart rate rise up to 200 BPMs. That number looks big when you know that’s your heart you’re clocking!
It felt great, but does anyone know if that’s excessive? I know I opened something up because when I stopped it felt like someone shoved a baseball bat down my lungs! I gagged like it was my first time (or maybe second time) doing aerobics work after a long period of idleness. I’m 40 years old, 6’2”, and around 215 pounds (the last time I checked my weight).
I’d appreciate any