I don't want to use a specific LiPo battery charging IC... If you think its possible can you guys give me some ideas?
Yes, it's possible. All you need is a voltage regulator with current limiting. You could do this with a regulator IC (switching or linear) which has a current limit feature, a 3 terminal regulator with a transistor to provide current limiting, op amp with Zener diode and external pass transistor etc., or a totally discrete design using only transistors and resistors.
However there are some provisos you need to be aware of:-
You must limit the voltage to 4.20 V maximum, with accuracy of +-0.03 V or better. Lower voltage is OK (it just won't get as much charge into the battery) but higher voltage will damage the battery - possibly causing it to catch fire and blow up!
Charging current must not exceed the maximum specified for your battery. Generally this is somewhere between 0.2 and 1 'C' (battery capacity in Ah).
If the battery is discharged below 3.0 V then the charging current should be reduced to ~0.1C or less until the voltage rises above 3.0 V. If your circuit cannot do this then you must not use it to charge the battery if it gets this low.
If the charger will be used in 'float' operation (continuously holding the battery at full charge) you should charge to a slightly lower voltage, eg. 4.10 V. Capacity will be reduced, but the battery will have longer lifespan.
For added safety the battery should have a PCM (Protection Circuit Module) to prevent overcharging etc. Some batteries have this built in. For those that don't you could add one to your charger circuit.
You could add all the features of a dedicated charger IC using generic parts, but the circuit would be quite complex. Sometimes this is worth it, especially if the battery is too small or large to suit available charger ICs.
A simple CVCC 'charger' may also be worth it if you don't mind doing to some work yourself (check the battery voltage before charging, disconnect it when charged, don't try to charge a battery which is not working properly or physically damaged), but is inherently more dangerous. That means you are responsible for making sure the rules are followed. Do not apply such a design to a commercial product or one that untrained people may use!