How strong is the "hydrophobic force"?
Hydrophobic interactions are weak interactions but can have greater strength than hydrogen bonds. I find the strength of the hydrogen bond in literature to be near 1 pN. Finding the strength of the "hydrophobic bond" has proven to be a bit more elusive. I would guess that there is a hydrophobic interaction strength spectrum (dipole strengths and such), but would also guess there is a fundamental order of magnitude of strength of the hydrophobic interaction.
Hydrophobic interactions result from the hydrophobic effect and manifest in measurable attractive forces. For example, the Polyhydroxyethylmethacrylate (pHEMA) backbone has dipole induced interactions, hydrophobic interactions, and hydrogen bonding; each of these interactions being identifiable in molecular position on the backbone (aliphatic regions being identified as hydrophobic sites).
To date, i have found a measurement of such resulting attractive forces in a specific interaction to be near 15 pN (oil-oil droplet AFM approach). However, I can also find literature that discusses forces in avidin-biotin complex which are dominated by hydrophobic interactions; resulting attractive forces range from 80 to 250 pN .
 Tabor et al, 2013; DOI: 10.1021/jz402068k
 Moy et al, 1994; DOI: 10.1126/science.7939660