I have no limits of space, transmit power or energy requirements so i
can use whatever i want.
Hardly. Bluetooth, which operates on 2.4 GHz frequencies is heavily regulated in regards to power output. Most of this legislation is expressed in ERP ("Effective Radiated Power") which means it will take antenna-gain and coax/feed losses into account in any particular direction or plain of radiation.
Furthermore, Bluetooth is classified in various power classes, Class 1 type equipment (which it must be certified to) has a maximum ERP of 100 mW (20dBm). Class 2 type equpment (usually mobile devices such as mobile/cell phones) are classified to 2.5 mW (4dBm).
If you want more power than that, I really think you need to be licensed for such, and your license will stipulate how much output power / ERP you may use.
I would strongly advise to check your local and country legislation.
I want to extend at "maximum possible" the range
There are various ways to do this, and staying within applicable legislation.
Assuming that you are building a "Class 1" type device and obtain the certification for such.
First, split the TX and RX signals to your device. You can do this with a "Circular Coaxial Coupler" for the intended 2.4 GHz frequencies (many references can be found by simply searching with your favorite search engine). This means that you can use two antenna's, one for TX and one for RX.
Second, you ensure that your TX power does not exceed the maximum allowable TX power of 100 mW ERP. You do need to know the following:
- Antenna Gain of the antenna you choose (A in dBi)
- Coaxial loss of the length of feeder you use (C in dB)
- Circular Coaxial Coupler insertion loss (CCC in dB)
With those you can now calculate the maximum TX power output of your amplifier you use:
Pout(dBm) = 20 - A + C + CCC
Lastly, you can use an very high gain antenna for RX; for omnidirectional RX you can use various "colinear vertical" designs, which could have typical gain figures of 9dBi or higher! You may consider vertically stacked array's of antenna's. Again you can use a search to obtain further info.
Tip: for both RX as TX antenna's, you can influence the angle of elevation of maximum gain by using stacked antenna arrays, fed with an different phase to influence "maximum gain takeoff angle", this to improve both RX as TX signal towards the horizon or horizontal plane, or even try to improve gain in negative angles below horizontal plane.
This [the need for phased arrays] you will have to establish by modelling your antenna system location, and the relative position of the users of the system. Example: if your antenna is really high (say 100 metres above ground) and your users are really low (say people walking on ground level using mobile phones) then you want a maximum gain of your antenna system with a negative angle of maximum gain, so maximum signal is downwards towards your users. Again, many designs for this, and info of such, can be found searching the internet.
Another thing to consider is: if your signal needs to travel through structures, such as buildings, walls, trees, or others, you may want to consider circular polarised antenna systems. Circular polarisation will improve penetration properties of the transmitted signal, at least to some extend.
In order to further improve range, you can consider directional antenna systems, but you have preempted this in your question, so I won't further comment on this.
Hopefully this will give you the right information to consider different antenna systems design. I have purposely used the term "antenna system" as this answer relates not simply to "one antenna, plug it in and play".
Afterall, you were asking for:
"maximum possible"
HTH.
