Air carrier Captain reported the aircraft would not pressurize using the Manual mode controller. Both the AUTO and STBY controllers were MEL 'ed out of service. The Captain stated the flight continued to the destination at 10;000 ft.

Narrative

Flight departed ZZZ at approximately XA25; en-route to ZZZ1 for a 52 minute Reposition; Non-Revenue Flight with Myself (Captain); First Officer; three Flight Attendants and one FME (Flight Mechanic.) Due to MEL XX-XX-XXX; we were limited to a maximum altitude of FL250 along with other restrictions per the MEL. We were cleared to FL250. While approaching FL200 in VNAV; the First Officer and I noticed that the Cabin Altitude was climbing too fast. He requested that I shallow the ascent which I did immediately by deselecting VNAV and engaging Vertical Speed; rolling the Vertical Speed Wheel to 250 feet per minute. Pressurization modes of AUTO and STBY were not operative (Reference: MEL XX-XX-XX) and the Pressurization Controller was moved to Manual. The First Officer manually closed the Outflow Valve; However; the Cabin Altitude continued to climb at which time; upon the cabin altitude reached 10;000 ft. and the Cabin Altitude Warning Horn annunciated. At that time; the First Officer panicked and reached for his Oxygen Mask and appeared to have great difficulty in donning it. Since I was the pilot flying; I immediately took manual control of the airplane and started my memory items of 'Cabin Altitude Warning' However; I was manually performing the rapid descent wanting to don my own Oxygen Mask but was unable to safely transfer controls to the First Officer as he was still fumbling with his Oxygen Mask. I simultaneously contacted ATC; informed them that we had a pressurization issue and we needed to descend. We were instantaneously granted descent to 10;000 ft. I attained the local Altimeter Setting; placed the Passenger Seat Belt Sign to ON; and placed both Engine Start Switches to Continuous. Since I was hand-flying the airplane; following the Memory Items of 'Emergency Descent' it was still not in the best interest of safety to transfer control as we were rapidly descending and I knew; being completely in control and 'on my own' without an experienced First Officer that we would be at the safe; breathable atmosphere in a matter of seconds. When the First Officer finally donned his Oxygen Mask; he failed to move his Audio Panel Switch from 'Boom' to 'Mask.' He started reading into his Mask from the QRH (Quick Response Handbook) and it was impossible to hear him. That being said; after nearly XX years at our company in Flight Training and Standards; I know the Boeing Procedures by heart and was fully in control of the situation. Upon reaching 10;000 ft.; the Cabin Altitude Warning Horn was still annunciation. I asked the First Officer to cancel the horn and he did not know how; so I reached up and canceled it myself. Once we were safely cruising at 10;000 ft.; Using my highly experienced skills in CRM; I conferred with my FME (Flight Mechanic) and my First Officer about returning to the departure field and we mutually agreed to continue the flight as the situation was now copacetic and the weather was extremely clear. The rest of the flight continued safely to an uneventful landing. Upon reaching the hard-stand at our maintenance facility; the ground mechanic pushed stairs up to the L1 Door. The Crew Van was waiting for us and my First Officer was the first one out of the airplane and was in the Van. I exited the Flight Deck and went to the first rows of seats where my three Flight Attendants and my FME were seated. I spoke to them to see if they were all okay. They all said they were fine. One of the new; young; Flight Attendants said she was scared and that her ears bothered her during the rapid descent. I told her that was normal and we all felt it. I got her assurance that she was Okay. We boarded the Van to the airport security gate; where a vehicle was waiting to transport the Flight Attendants and the First Officer to the hotel. I had a friend waiting there for me in his vehicle so I did not ride with the three Flight Attendants and the First Officer. Later; without my knowledge; the First Officer convinced the young Flight Attendant that she 'had to go to the hospital' to have her ears checked out. I understand she did and was medically cleared; but due to her naive and inexperience; I was told she attained a week off to 'recover.' I was flabbergasted that my First Officer actually convinced the girl that she had a problem and 'needed medical attention.' The thought that a member of my Crew sought medical attention without my knowledge was very disturbing to me. Had I known that; I would have accompanied her to the hospital. After over 47 years as a Professional Aviator; nearing 18;000 hours as a pilot with an additional XX;000 hours teaching pilots in Level 3 Simulators at the training center; I did the very best I could under the circumstances to Aviate; Navigate and Communicate to the best of my ability.