Find whether one mean is very different from all others in multiple measurements

#1
I have many data frames like the one reproduced below in R. I am trying to find cases where one group only is very different from the other groups (I have a variable number of groups for the different data frames). I have tried with ANOVA, but this test only tells me if the group means are different from each other, and almost all are, but what I am looking for is either a test or a measure that can point to whether there is one group mean that is significantly different from the others, something like TukeyHSD but I don't think this works because I need to account for the repeated measurements on the same individuals and I cannot assume a linear model (so I think also mixed models are not appropriate).

This is my data:
df = structure(list(groups = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L), .Label = c("group1", "group2", "group3"), class = "factor"), individuals = structure(c(1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L), .Label = c("sample1", "sample2", "sample3", "sample4", "sample5", "sample6", "sample7", "sample8", "sample9", "sample10", "sample11", "sample12", "sample13", "sample14", "sample15", "sample16", "sample17", "sample18", "sample19", "sample20", "sample21", "sample22", "sample23", "sample24", "sample25", "sample26", "sample27", "sample28", "sample29", "sample30", "sample31", "sample32", "sample33", "sample34"), class = "factor"), value = c(0, 0.126889234269648, 0.0568617212177722, 0.0655431535336275, 0, 0, 0.0567384158931187, 0.0109376655462734, 0, 0.0948086895930572, 0, 0, 0.0824709304159648, 0, 0, 0, 0, 0, 0.033917835818341, 0.0198035710833954, 0.133552284941515, 0.0664567537955113, 0.0430308231270825, 0.0160783942164045, 0.0738182676544177, 0.120422772617204, 0.135233112827437, 0.0860623347815552, 0.0200020675534846, 0, 0.0973088048201884, 0, 0, 0.038283321330165, 0.225376778959931, 0.137536577951645, 0.426342384989027, 0, 0.337720200488891, 0.384254666977513, 0.0108438100713879, 0.278754009620225, 0.381097277894091, 0.154981762820021, 0.0864810371628134, 0.218313535529244, 0.202077489447883, 0.117036020922844, 0.129779767219537, 0, 0.559959304303877, 0.240102470306958, 0.252798090888197, 0.100823988780093, 0.126391112137011, 0.108138613814024, 0.0692924355684719, 0.459841817208745, 0.138288519068394, 0.113182370152051, 0.0525657185927877, 0.177169361598713, 0.355934492361351, 0.186605323013182, 0.148750791970728, 0, 0.155418848968847, 0.0600295032575727, 0.774623221040069, 0.735574187778708, 0.5167958937932, 0.934456846466373, 0.662279799511109, 0.615745333022487, 0.932417774035493, 0.710308324833502, 0.618902722105909, 0.750209547586922, 0.913518962837187, 0.781686464470756, 0.715451580136152, 0.882963979077156, 0.870220232780463, 1, 0.440040695696123, 0.759897529693041, 0.713284073293462, 0.879372440136512, 0.740056602921473, 0.825404632390465, 0.887676741304446, 0.524079788574851, 0.787893213277188, 0.766394857230745, 0.812201168579775, 0.736768303619732, 0.624063440085165, 0.813394676986818, 0.720479745676339, 1, 0.786220846027069, 0.901687175412262)), .Names = c("groups", "individuals", "value"), row.names = c(2L, 6L, 10L, 14L, 18L, 22L, 26L, 30L, 34L, 38L, 42L, 46L, 50L, 54L, 58L, 62L, 66L, 70L, 74L, 78L, 82L, 86L, 90L, 94L, 98L, 102L, 106L, 110L, 114L, 118L, 122L, 126L, 130L, 134L, 1L, 5L, 9L, 13L, 17L, 21L, 25L, 29L, 33L, 37L, 41L, 45L, 49L, 53L, 57L, 61L, 65L, 69L, 73L, 77L, 81L, 85L, 89L, 93L, 97L, 101L, 105L, 109L, 113L, 117L, 121L, 125L, 129L, 133L, 3L, 7L, 11L, 15L, 19L, 23L, 27L, 31L, 35L, 39L, 43L, 47L, 51L, 55L, 59L, 63L, 67L, 71L, 75L, 79L, 83L, 87L, 91L, 95L, 99L, 103L, 107L, 111L, 115L, 119L, 123L, 127L, 131L, 135L), class = "data.frame")


If I just use this TukeyHSD(aov(value~groups, data=df)) the p-values are significant for group2 against 3, which is not what the plot shows: plot(value ~ groups, data=df)...

Any help/suggestion is very much appreciated.
Thank you in advance,
Francy