Highest Pressure Human Body Can Survive In?
The deepest recorded simulated dive is still the on-shore French Hydra 10 experiment at 701 msw (meters salt water); at a pressure of 71 atm. Since breathing a mixture containing helium is required at such depths (to avoid the extreme narcosis produced by nitrogen at such depths), the new problem helium causes is


  High-pressure nervous syndrome (HPNS). HPNS, brought on by breathing helium under extreme pressure causes tremors, myoclonic jerking, somnolence, EEG changes, visual disturbance, nausea, dizziness, and decreased mental performance. Symptoms of HPNS are exacerbated by rapid compression, a feature common to ultra-deep "bounce" dives.


Actually that Wikipedia description is a bit misleading, the problem is caused by any gas mixture, not just helium, but other gasses like nitrogen or hydrogen have a narcotic effect that counteracts HPNS. HPNS sets in at about 120m:


  The high pressure
  neurological syndrome (HPNS) begins to show signs at about 1.3 MPa (120 m) and its effects intensify at greater depths.
  HPNS starts with tremor at the distal extremities, nausea, or moderate psychomotor and cognitive disturbances. More severe
  consequences are proximal tremor, vomit, hyperreflexia, sleepiness, and psychomotor or cognitive compromise. Fasciculations
  and myoclonia may occur during severe HPNS. Extreme cases may show psychosis bouts, and focalized or generalized
  convulsive seizures. Electrophysiological studies during HPNS display an EEG characterized by reduction of high frequency
  activity (alpha and beta waves) and increased slow activity, modification of evoked potentials of various modalities (auditory,
  visual, somatosensory), reduced nerve conduction velocity and changes in latency. Studies using experimental animals have
  shown that these signs and symptoms are progressive and directly dependent on the pressure. HPNS features at neuronal and
  network levels are depression of synaptic transmission and paradoxical hyperexcitability. 


The problem with extreme helim pressure was partially alleviated in the Hydra experiments by mixing/replacing some of it with hydrogen, in the so-called Hydreliox mixture.


  For the Hydra VIII mission at 50 atmospheres of ambient pressure, the mixture used was 49% hydrogen, 50.2% helium, and 0.8% oxygen.


The French did this based on prior experiments which found that the narcotic effect of hydrogen reduces HPNS (or even completely eliminates it at lower depths):


  A H2-He-O2 mixture with 54 to 56% hydrogen was studied with 6 subjects (professional divers) during 2 dives to 450 m. The 38-h compression was the same as that used with other types of breathing mixtures (He-O2 and He-N2-O2). The results obtained during compression and during the stay at 450 m in H2-He-O2 show that the EEG changes (increase of theta activities in the anterior regions of the skull, decrease of alpha activities) are similar to those found with other respiratory mixtures. On the other hand, the other symptoms of high pressure neurologic syndrome (HPNS) were clearly improved for the same depths. Thus, neurologic symptoms (tremor, dysmetria, myoclonia, drowsiness) are nonexistent, and the performances during psychometric tests remain similar to those of the surface. Hydrogen, with its narcotic potency, suppresses some symptoms of HPNS and seems to open new perspectives for deep diving.


Of course, having hydrogen (in such large proportion) and oxygen in a gas mixture poses dangers of fire, explosion etc. This is avoided by decreasing the oxygen concentration; alas this cannot be a one-step process (for diving):


  The major problem with hydrogen-oxygen mixtures is the potential for explosion. Although the concentration of oxygen needed for combustion of oxygen-hydrogen mixes varies a bit with pressure, a general rule of thumb is that hydrogen-oxygen mixes above 5 %  O2 are at-risk.. So, to avoid nasty fires and explosions, hydrogen is only considered as a breathing gas component at pressures where a less-than 5% oxygen concentration in the breathing gas mix gives a partial pressure of oxygen great enough to sustain life. Perhaps the most common example of a hydrogen-fire related disaster is the destruction of the Hindenburg dirigible. [...]
  
  In 1944 Arne Zetterstrom discovered a way to breach the transition between compressed air and Hydrox without risking explosion. The technique was to descend to 100 feet and switch to a 4% oxygen / 96 % nitrogen mixture. After breathing this mix for sufficient time to allow the oxygen concentration in the lungs to drop below the "explosion threshold," the diver switched to Hydrox and continued descent. On ascent, the diver again used the Nitrox (4% O2 / 96% N2) as a transition between Hydrox and air. Using this technique, he descended to 363 feet. At that depth, the alteration in voice characteristics, coupled with excitement, made communication impossible and additional dives used a telegraph key.  [...]
  
  During the mid 1960's research into the use of hydrogen in breathing gases resumed with animals breathing Hydrox for up to 24 hours at 70 Ata. One interesting aspect of the animal research was the suggestion that hydrogen reduced the HPNS (high pressure nervous syndrome) often observed with helium based gas mixes on deep dives. Ultimately animals would be taken to 3500 feet on hydrox. 
  
  In 1983 COMEX, the French deep diving concern (perhaps more famous in the US as the company providing the submersible used in the recovery of artifacts from the Titanic) began a series of dives to investigate the narcotic potential of hydrogen. Divers including H.G. DeLauze, President of COMEX, descended in open sea to approximately 300 feet for five minutes. The divers could not perceive a difference between Hydox and Heliox at that depth.  Chamber dives to 300 m (984 ft) demonstrated that hydrogen possessed a narcotic effect different from nitrogen. Hydrogen narcosis (the "hydrogen effect") had a tendency to be more psychotropic, i.e. more like LSD, while nitrogen narcosis had an effect similar to alcohol. This deeper work suggested that Hydrox as a binary gas mix would not be too useful at depths below about 500 feet. 


Based on that COMEX developed a protocol of switching from heliox to hydreliox at 250msw.

A more recent (1994) COMEX paper reported that at 500m (on hydreliox), manual dexterity was about 80% of the surface one, while the arithmetic ability decreased to 60%. So not dying and being able to function as on the surface, aren't the same thing.

Also these experiments are time consuming and costly. In Hydra VIII, the whole thing took one month, of which only 10 days or so were spent at maximum depth. And in case "something bad happens" interventions aren't easy in such settings.

A 1984 American experiment with trimix at 650m was aborted after one of the three subjects (despite negative previous neuropsychiatric screening, as well having substantial commercial diving experience and successfully taking part in the previous experiments of the series) developed hallucinations and then full blown mania at 625-650m.


  The Atlantis IV dive was designed to test the ability of 5%
  N2 trimix breathing mixture (5% N2, 0.5 atmosphere
  oxygen, with the balance helium) to counteract the
  [high pressure nervous] syndrome without nitrogen narcosis. [...]
  
  The dive began with an initial compression rate of
  30 m/hour to 300 m (1000 ft) sea water equivalent
  pressure. By day 3 of the dive, at about 470 m, subject
  C had developed significant insomnia. On day 4 (540
  m), this subject became stressed and complained of
  mild auditory illusions of music and visual distortions
  consisting of a “halo effect” around the chamber’s
  apparatus. On day 5 (625 m) of the dive, the subject’s
  irritability, insomnia, and agitation worsened. He was medicated with temazepam, 60 mg, and achieved
  substantial relief of these symptoms after 15 hours’
  sleep. However, with further compression to 650 m,
  subject C experienced a recrudescence of auditory and
  visual illusions and developed rapid speech, racing
  thoughts, shortened attention span, and difficulty in
  focusing on tasks. Ankle clonus, marked hyperreflexia,
  and myoclonic jerking were documented, and he complained of nausea and weakness. In contrast, the other
  two subjects performed normally in almost all respects.
  On day 9 (650 m), the subject stated he felt “paranoid”
  and that he felt like “I’m going insane.” He
  began to carry a mirror to look at himself to “remind
  myself that I’m not a raving maniac.” He stated that
  colors were enhanced and vivid and he could see
  “unusual” mosaic patterns on the metalwork within
  the chamber. He was unable to comply with instructions
  to enable cognitive function to be tested because
  of extreme agitation and distractibility.[...]
  
  Early on day 9 (650 m), the decision was made to
  halt the dive, but decompression from 650 m is still
  experimental and could not be accelerated. Because the
  decompression process was expected to last as long as 30 days, it was necessary to consider medicating the
  subject more intensively. Since lithium had been shown
  to exacerbate symptoms of the high pressure nervous
  syndrome in animals and the effects of phenothiazines
  at such high pressures were virtually unknown,
  the decision was made to control the diver’s agitation
  with benzodiazepines. Accordingly, diazepam was used in doses up to 120 mg/day. This treatment only
  mildly sedated the subject and provided no real control
  of his behavior.
  The subject’s agitation persisted over the next 14
  days and was marked by decrements in his cognitive
  function and bursts of aggressive paranoia, grandiosity,
  and irritability. By day 24 of the dive (383 m), the
  subject’s irritable and aggressive behavior was becoming
  potentially dangerous and led to the decision to
  medicate with chlorpromazine, up to 300 mg/day.
  Within 48 hours of the beginning of chlorpromazine
  treatment there was dramatic improvement in the
  subject’s behavior. He became relatively calm and
  better organized. He denied having illusions or hallucinations. [...]
  
  Ten days after exiting the chamber, the subject again
  exhibited signs of hypomania including hyperactivity,
  insomnia, and irritability. Lithium carbonate therapy
  lessened these symptoms and was continued prophylactically
  for 6 months. It was then discontinued
  without any recurrence of symptoms.