Look closely at the top plot and you can see it is just applying a simple maximum current OR voltage scheme. Up to about .9 hour the maximum current (looks like about 700 mA) causes lower than the maximum voltage, so the battery is charged with that current. When the battery reaches the 4.2 V limit, the charger holds the voltage there and the current then decreases. Eventually the battery gets so full that the charging current tapers off to nearly 0.
The charger puts out either 700 mA or 4.2 V, whichever results in lower voltage (or current). The battery then decides the other of the two.
Q: What is the reason Lithium Polymer batteries are limited to 1C charging?
A: They're not, at least not fundamentally. You can get 5C and 10C charging batteries from e.g. A123 systems. The reason batteries are limited on paper to relatively slow charging rates is to avoid internal hot spots, caused by a chemical-physical phenomenon that causes lithium chemistry batteries to discharge relatively evenly over their electrodes, but actually concentrates charging current to the areas that have the highest ion mobility. This is chemistry dependent; there are cathode materials that avoid this problem mostly and thus can charge faster. It's fairly specialty stuff though, and it is generally not a good idea to use chemical energy storage for rapid-cycle applications. I will come back to this later
Q: If I'm charging a Li+ battery for 90s at 1C, can I safely charge at more than 1C?
A: Most Li+ chemistries can safely charge up to 2C with a voltage cap, but with reduced lifetime. Not deeply discharging the batteries will alleviate this though, and then some. The cycle life of a battery increases at least tenfold if you discharge it at most 10% instead of 50%. Check with the manufacturer or datasheet to make sure though. If there is no datasheet, you shouldn't use the battery in a production environment anyway.
Addendum:
It's not a good idea to use a battery as a quick-cycle energy storage. Even with shallow discharging, especially high current batteries will only last you about 10 000 cycles. This is what (ultra)capacitors are for, especially if you are oversizing the battery anyway. The only downside to a capacitor is the ~10x decrease in volumetric efficiency compared to a battery. Without that problem, capacitors are much more suitable for quick charging and discharging. You can charge Maxwell's BCAP series within 15s, for instance. That's 1000 and 3000F capacitors, hundreds of amps. Besides, with large charging and discharging currents, your supporting circuitry and cabling will probably be larger than the power source anyway.
Also, it's generally a much better idea to store and use lithium chemistry batteries at the top of their capacity, near 4.25V. You get more power out of them for less current and you charge them faster with lower C-rating. Also their apparent internal impedance is much lower the 'fuller' a Li+ battery is.
Best Answer
The battery concerned is LiIon or LiPo.
This is relevant to the question. I'll say LiIon below - it applies equally to LiPo.
Short answer: Charging at somewhat below max allowed rate may cause a modest improvement in battery cycle life but it will not be a major factor.
Longer:
A LiIon battery has a max allowed charge rate that is specified by the battery maker.
This is usually the C/1 rate = a current in mA equal to the mAh capacity of the battery.
IF you charged at the C/1 rate for one hour the mAh transferred would equal the mAh nominally available from a fully charged battery.
However, this is not how LiIon chemistry works.
From flat you charge at Imax or less (usually C/1 or less) until V_charge_max is reached (typically 4.2V per series cell) and THEn the charger holds Vcharge at V_charge_max and the battery accepts a declining current with time that depends on battery chemistry. When I charge falls to I_charge_min charging is complete and Vcharge MUST be removed. Floating a LiIon battery at V_charge_max tends to destroy it rapidly.
In your case the 85W charger is probably sized to charge at I_Charge_max when the charging circuitry allows it to. The 45W charger max current is below I_charge_max, so until V_charge_max is reached charging occurs at I_best_45W_supply_can_manage.
This reduced charge rate MAY increase battery lifetime but probably not appreciably as while many other factors are discussed and documented this is not. You cannot charge the battery "too fast" by using a higher wattage power supply as the actual charger circuitry is internal to the laptop and it will limit Ichg to C/1 or whatever rate the battery is rated at.
Lower Ichg reduces heating, probably allows somewhat more even deposition of Lithium in the battery and is probably therefore less mechanically wearing on the battery. mA major life affecting factor of LiIon batteries is that they are mechanically "pumped" by the charge/discharge action. The batteries beat themselves to death. [This is a major factor in the long life of LiFePO4 batteries - an internal Olivine structure provides mechanical support for the battery structure - and takes up room so reduces battery capacity.]